Recently I wrote about the odd given name "Chasity." This brought a response from SQ that "I'm sure Chasity is uniquification of Chastity." I think "uniquification" is exactly right. These parents believe that it is possible to give their child a unique name. Moreover, they think it would be a good thing for their child to have a unique name.
Years ago the British sociologist Basil Bernstein studied the speaking styles of working class and middle class Londoners. He noticed that they were likely to use different ways of speaking, which he called the restricted code and the elaborated code, respectively. The restricted code is best used when all the people in the conversation know one another and have many shared understandings. Sentences tend to be concrete and specific. The elaborated code, on the other hand, is more abstract and general, filling in points that might be obscure to people not in the speaker's immediate circle.
Bernstein thought that working class speakers used a restricted code most of the time, even when talking to strangers, because most of their social life was spent with people they knew talking about things they all knew. Middle class people, on the other hand, often used the elaborated code even with their intimates. Each code is an effective way of speaking in the right circumstances, and each can be mysterious or cumbersome in the wrong circumstances.
I think working class parents are much more likely to try to give their children unique names than middle class people are. If you think primarily in terms of your social circle, then it is easy to imagine coming up with a unique name. You are also likely to refer to people only by their first names in your group of regulars, so one fancy name would do. I think this is why rappers often go by one name.
If, on the other hand, you think that your child will live in unknown places in the great world, then you are more likely to see how unlikely it is that you can come up with a unique name. Your family name is more likely to be part of how you are known and referred to.
I think the middle class also avoids uniquification of their children's names because weird names and, especially, weird spellings (or, in the case of Chasity, misspellings) of common names is itself a sign of lower class origins.
In a middle class world, William is a dignified, if formal name, as ordinary as a kitchen clock. Weeyum, [which I hope I made up] on the other hand, suggests parents who were thinking more about yelling the name in the neighborhood than how it would appear on school and job applications.