Back in the old old days, the average person had a single name. Later, after the population grew, it became necessary to distinguish Tom from Tom and Dick from Dick and Harry from Harry. So we got Tom Jones and Tom Swift, and so on.
Later, the population grew some more, and it is now common for
people to have a third (middle) name. And in some places people have even more. I once met an Egyptian fellow whose name seemed to consist of all the first names of his last 30 male ancestors.
That's probably a bit extreme, although this IS the "Cultural" section of the HalfBakery, and so it doesn't hurt to understand why a particular culture might do something like that. The answer is probably closely related to the Western practice of using "family" names.
There is something about "direct male line of descent" that various male-dominated cultures have this "thing" about.
But, hey, this is the 21st Century! Equality is an important concept. Women contribute as many genes as men to offspring (more, actually, since the X chromosome is considerably bigger than the Y).
And while the Y chromosome can be a valid way to trace direct male line of descent, the mitochondria (subcells within ordinary cells) are an equally valid way to trace direct female line of descent.
That means we can logically have female-related family names, just as we mostly now have male-related family names.
So why not both? Population is still increasing, and at some point it would be useful to distinguish the average person by assigning four names at birth.
Anyone who wants to bollux modern bureaucratic computerized systems, that are only designed to handle 3 names per person, might like this Idea....
Let's pretend you want to adopt this system. You should start with as thorough a Family Tree as can be obtained. That's because you want to know as much about your female ancestry as your male ancestry. While your current family name is PROBABLY the name of your father's father's father's father's..., you need to know the family name of your mothers' mother's mother's mother's....
Then you get your name legally changed, to "fit" this dual-lineage convention. Assuming you already have an ordinary middle name, you would keep that and your first name (although there's nothing stopping you from changing them, too, if you wish). The generic format is:
First name
"Middle" name
Family name of opposite-sex parent
Family name of same-sex parent
I'll create a couple of examples: Alan James Smith Wilson and Susan Nichole Bourne Thompson. Names are kept throughout life, and are not changed for silly reasons like marriage. So, suppose these two get married and have kids. The last two names of any boys would be Thompson Wilson, and the last two names of any girls would be Wilson Thompson.
Thus do the boys carry on a typical direct male line of descent, in a naming convention, while the girls can now also carry on a direct female line of descent, in an equivalent naming convention.
It may be odd-looking and unconventional, but so what? It can WORK. It's fair. And people who grow up with it will have no trouble being used to it, no matter what any male egos, dominating their culture, have to say about it.