[unparsed]
I liked pattern Two best because
(1) every child has or had grandparents;
(2) it's very likely that the child's grandparents have or had oldest brothers and/or oldest sisters;
(3) there's an increased chance that one or more of the grandparents' oldest brother and/or oldest sisters is deceased before the child's "christening", or will be by the time the child is of age.

Pattern One seems to me to depend too much on chance; the child's parents are only about 50% likely to have third-oldest brothers or third-oldest sisters, even if the child's grandparental couples did have seven children each. And there's a fairly good chance (though less than 50%, if the grandparents had six or seven children) that one parent doesn't have a second-oldest brother or doesn't have a second-oldest sister.
Also, the child's grandparents may still be alive; so may the child's parents' siblings.

I think maybe I should have picked Pattern Three instead of Pattern Two.
I didn't, because
(1) I thought some ancestor's siblings should have somebody named after them and
(2) remembering who the child's great-great-grandparents were may be too much of a strain on the parents' memories (those would be the parents' great-grandparents; odds are they died when the parents were young children, if not before that). This is especially true in a pre-Industrial, pre-Enlightenment society, when the parents' grandparents probably also have died well before the child is born, and are not too unlikely to have died while the parents were still quite young.

Pattern One is grandfathers or grandmothers, then uncles or aunts;
Pattern Two is great-grandfathers or great-grandmothers, then granduncles or grandaunts;
Pattern Three is great-great-grandfathers or great-great-grandmothers, then great-granduncles or great-grandaunts (although you'd need to have nine or more children of the same sex to get past the great-great-grandparents).

So Pattern One does in fact already go to grandparents; in fact it goes to them first.

All three patterns result in cycles of names being used in lines of fathers-to-firstborn-sons and mothers-to-firstborn-daughters; a cycle of two names in Pattern One, a cycle of three names in Pattern Two, and a cycle of four names in Pattern Three.

Pattern One is designed to keep a child from being named after that child's parent; but it results in the names of a line of parents-to-firstborn-children-of-the-same-sex, alternating. Father-to-firstborn-son names alternate; mother-to-firstborn-daughter names also alternate.

Pattern Two is designed to keep a child from being named after that child's parent or grandparents. Father-to-firstborn-son names cycle through a set of three names; mother-to-firstborn-daughter names also cycle through a set of three names.

Pattern Three is designed to keep a child from being named after that child's parent or grandparents or great-grandparents. Father-to-firstborn-son names cycle through a set of four names; mother-to-firstborn-daughter names also cycle through a set of four names.

In Pattern One, your father's oldest brother will be named either after your father's father's father (if he was your father's parents' first-born son) or your father's mother's father (if it was your own father, instead, who was your father's parents' first-born son); either way, a great-grandfather. So a third-born son will be named after a great-grandfather, one of his father's grandfathers.
And your mother's oldest brother will be named after your mother's father's father; another great-grandfather. So a fourth-born son will be named after a great-grandfather, one of his mother's grandfathers.
And your father's second-oldest brother will be named either after your father's mother's father (if he was your father's parents' second-born son) or your father's father's oldest brother (if your own father who was either the first- or second-born son to his parents); one way a great-grandfather, the other way a granduncle. So a fifth-born son will be named either after one of his great-grandfathers (one of his mother's grandfathers) or after one of his granduncles (one of his father's uncles).
And so on.
Similar statements apply to third-and-subsequent daughters.

And similar reasoning applies to Pattern Two for fifth-and-subsequent sons and daughters; for instance, for a fifth-born son, the boy's father's father's oldest brother has been named either after the boy's father's father's father's father or the boy's father's father's mother's father, depending on the birth-order of the boy's father's father in the father's father's parents' family.

So it's just as true to think of Pattern One as going from grandparents to great-grandparents to great-great-grandparents etc. (and Pattern Two going from great-grandparents on back, and Pattern Three going from great-great-grandparents on back), as it is to think of Pattern One as going through the uncles/aunts in descending order by age (and Pattern Two going through the granduncles/grandaunts, and Pattern Three going through the great-granduncles/great-grandaunts).

Even back in hard times, Pattern One probably would result in many children being named after living relatives.
Before the Scientific Revolution (or even later), Pattern Two would have meant most children's namesake relatives were deceased by the time the child was named; but in modern times, it would have modern children's namesake relatives still be living when the child is named.
Even in modern times, Pattern Three probably would result in most children's namesake relatives being deceased by the time the child was named.


I wonder about that and haven't been able to figure it out -- can you?

Pattern One is actually used on some Greek island; if you've ever been a guest of a Greek extended family you may have noticed how common duplication of names is.

If the culturally appropriate thing to do when there is not nor has there ever been a relative of the prescribed sort after whom to name the child*, allows the possibility of just making up a new name or of naming the child after a non-relative -- maybe even a foreigner -- , then new names may enter the system. In that case the problem you mention might get solved.
But if the thing to do when no relative of the prescribed sort has ever existed, is to name the child after some other relative, then this kind of fix is unavailable, or at least won't be used much.
*(In Pattern One, not likely unless there are third-or-later sons or third-or-later daughters. In Pattern Two, not likely unless there are fifth-or-later sons or fifth-or-later daughters. In Pattern Three, not likely unless there are ninth-or-later sons or ninth-or-later daughters.)

Also, I expect whichever naming-pattern is used, to be a cultural habit or tradition for which there is cultural pressure to comply, but for which there is no legal sanction to punish non-compliance. Some couples of parents won't always comply for every child because of ignorance (e.g. they don't know the name of the prescribed relative); some because of bad feeling toward the designated relative; some for various other reasons, for instance particularly loving a name other than the one designated. If so, there might be a trickle of new names into the system, and/or some of the later names that you would expect to die out with fewer children wouldn't die out as fast as you might expect them to.

Notice also how second- and fourth- and sixth-born sons bring in names from their mother's family, and 2nd- & 4th- & 6th-born daughters bring in names from their father's family. So, techically, if the families inter-marry pretty mixedly, then as long as the population is increasing (or, at any rate, as long as both the male and female populations are increasing), however slowly, the names are not likely to die out; not even if people start having children at barely more than the replacement rate. If a particular patriline is running low on male names it will import some from the lines of the females it marries; and sim for matrilines and female names.

However if the population is stable (or, worse, declining), then yes, names will die out. Then once the population begins to grow again, there may be a smaller pool of names to draw from, unless someone remembers or has recorded the names of the relatives who lived before the population quit growing or started to decline.

Or, at least, that's my guess; but like I said I haven't been able to prove it.
Does anyone have better ideas?

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@Blake; BTW did you post your own concultures' naming systems on the CWBB somewhere?

In fact there are European countries in which a child is always named after the saint upon whose day the child was born. Related children thus rarely have the same name; nor do children often share a name with someone from an earlier generation of their own family. But each one shares the name with over 0.27% of their compatriots of the same sex. It's just that those probably don't include any, or at least not more than one or a few, of their close relatives or classmates.

In more than one East Asian culture, each family has a cycle of names -- typically three or five, sometimes seven, or some other number -- and each child in any generation receives as one name the next name in that cycle.
For a Chinese family I know, the name from the eight-name cycle is always the first part of the given name. The grandfather's generation were all Chang Wei-something (e.g. Chang Wei-Ding); the father's generation are all Chang Wan-something (e.g. Chang Wan-An, Chang Wan-Jing, Chang Wan-Ling); and the children's generation are all Chang Ho-something (e.g. Chang Ho-Ming).
In Korean cultures, the cycles are of odd length, and generations alternate whether the name from the cycle is the first part or the second part of the given or personal name. For instance, suppose the Smiths (a Korean family) have the three-name cycle Abraham Barnard Charles. Then maybe Smith Abraham-David's son is Smith Edward-Barnard, whose son is Smith Charles-Frederick, whose son is Smith George-Abraham, whose son is Smith Barnard-Herbert, whose son is Smith Isaac-Charles, whose son is Smith Abraham-Jonathon.
I understand Vietnamese culture is similar when it comes to naming.
I've been told by a Chinese family from Vietnam that this means that if you meet someone with your same surname who was born in the same county you were born in, that you can tell by knowing their full name, how they are related to you.

Some cultures name babies strictly according to birth-order within sex. IIUC Dakota couples always named their first-born daughter Wynona. There'd be even more duplication of names among such a culture than there is in the patterns I described in my opening post.

I think I should try it with just 20 to 40 founding couples.

The reason I have it all written out in the PM I sent you, and so many founding couples, is exactly that I don't have a reliable way to make random things happen randomly; so I tried to make everything that could happen, happen, in proportion to its chance of happening.

It would be easier to write and run a program. I'll have to find out whether I can.

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I finished my fifth simulation of Naming Pattern 1. (The first four attempts had various flaws.)

I'll try to share the results as graphs, when I can figure out how.

Meantime here are the main results.

I started with 26 masculine names, the capital letters A thru Z; and 26 feminine names, the lowercase letters a thru z.

In 40 generations no name disappeared entirely. (I figured a millenium would be about 40 or 50 or so generations.)

Here are the ratios of the least common to the most common names, as percentages;

Generation   Masculine Names   Feminine Names
 1st gen     100.00%           100.00%
 2nd gen      59.38%            58.06%
 3rd gen      54.55%            64.52%
 4th gen      50.00%            45.45%
 5th gen      48.57%            48.48%
 6th gen      41.67%            35.29%
 7th gen      30.56%            44.12%
 8th gen      38.46%            31.58%
 9th gen      32.43%            37.14%
10th gen      30.23%            19.51%
11th gen      34.21%            34.29%
12th gen      20.83%            13.33%
13th gen      37.21%            30.56%
14th gen      13.73%            18.60%
15th gen      26.00%            18.42%
16th gen      10.71%            21.74%
17th gen      20.83%            17.02%
18th gen      10.53%            19.15%
19th gen      17.65%            17.02%
20th gen       7.02%            25.00%
21st gen       9.09%            20.41%
22nd gen       9.62%            21.95%
23rd gen       5.17%            13.73%
24th gen      12.00%            26.19%
25th gen       4.55%            17.39%
26th gen      11.67%            30.00%
27th gen       5.00%            16.00%
28th gen      10.17%            18.75%
29th gen       3.08%            14.29%
30th gen      12.90%            16.33%
31st gen       4.62%            12.96%
32nd gen      13.33%            13.64%
33rd gen       4.76%            19.61%
34th gen      16.42%            13.33%
35th gen       6.06%            17.54%
36th gen      10.00%            16.67%
37th gen       6.76%            18.18%
38th gen      10.77%            14.81%
39th gen       4.62%            18.97%
40th gen       8.22%             9.43%

As you can see, within a general downward trend, the ratios seem to almost alternate between increasing and decreasing; but mostly each decrease is a little larger than the previous increase.
The lowest figure in the male column is the 4.62% in Generation 39.
The lowest figure in the female column is the 9.43% in Generation 40.

I started in Generation 1 with 650 (=26*25) couples having 650 sons and 650 daughters. A husband of each masculine name had a son of each other masculine name; and a wife of each feminine name had a daughter of each other feminine name.

After that, in each generation, the men each randomly married one woman and the women each randomly married one man.
It wasn't completely random; to prevent incest and to prevent certain problems picking names, a wife's father couldn't have the same name as her husband or her husband's father or her husband's oldest brother (if he had one) or her husband's mother's father; and a husband's mother couldn't have the same name as his wife or his wife's mother or his wife's oldest sister (if she had one) or his wife's father's mother.
But, other than that, it was random.

In each generation after the first, the total number of sons exceeded the number of married couples by one; the total number of daughters also exceeded the number of married couples by one.
Couples might have from zero to three children.
Up to one couple (randomly chosen) would have no child.
Two to four randomly chosen couples would have only one child; one to two would have just one son and no daughter, and one to two would have just one daughter and no son.
Six to eight randomly chosen couples would have three children. One randomly chosen couple would have three sons and no daughter; one randomly chosen couple would have three daughters and no son; two to three randomly chosen couples would have two sons and one daughter; and two to three randomly chosen couples would have two daughters and one son.
Of the remaining couples, all of whom would have exactly two children; a randomly-chosen quarter of them would have two sons and no daughter; a randomly-chosen quarter of them would have two daughters and no son; and a randomly-chosen half of them would have just one son and just one daughter.

First sons were named after the husband's father; second sons were named after the wife's father; third sons were named after the husband's oldest brother if he had one, else after the wife's oldest brother if she had one, else after the husband's mother's father.
First daughters were named after the wife's mother; second daughters were named after the husband's mother; third daughters were named after the wife's oldest sister if she had one, else after the husband's oldest sister if he had one, else after the wife's father's mother.

In each generation there'd be exactly one third son and exactly one third daughter.

In each generation there'd be one or two husbands and one or two wives with no siblings at all; whether or not one such husband married one such wife, and whether or not they had a third son or a third daughter, were determined by chance, but the odds against it were very high (I guess about 650^3 to 1).

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I will try to graph the percentages above.
I'll also try to graph the numbers of sons of each name, and the numbers of daughters of each name, against the generation-number.
If I can post those here, I'll also try to make my detailed data available for folks to look at; I don't think I should post the births and marriages registers for 650 to 689 couples for 40 generations.

@Riven:
This is the kind of thing I’d like to know how to do!