This is something which developed over the last 40 years as the creation of brand new names for children became a strong part of African-American culture.
I don't know of formal research on this. However, I believe that the idea of having apostrophes as part of the spelling of names comes from three sources.
First, it comes partly from the existence of French, Italian, and Irish surnames that do use the apostrophe in their spelling. D'Amato, L'Hereux, O'Connor, etc. are just a few of the many examples of such surnames which occur in the United States. D'Andre, which was one of the first of these names to become popular in the African-American community, does exist as a French surname in that spelling. I don't think this goes back to the "French being sympathetic to abolishing slavery"; I think it goes back to a combination of many African-Americans having ancestry from Louisiana, where French surnames are common, and from the general American idea that anything French looking or sounding is "elegant".
The other large influence comes simply from a problem with the lack of use of diacritical marks in American typography. In most other European languages, there are certain accent marks that are used in the spelling of many words and names. Among the most common are the marks seen in "é" and "è". Before the last few years, most typescripts used in the United States simply did not have such diacritical marks available. However, they are very common in French and Spanish names. Many parents who were attracted to such names because of their idea that they were sophisticated or elegant wanted to reproduce the marks in names such as ANDRÉ or INÈS. It was impossible, however, to put such marks on a birth certificate or other official forms. The closest you could get was to use an apostrophe instead. A great many of the apostrophes that occured in African-American names back in the 1960s when they first became common are placed in such a way that I'm sure they were "standing in" for the diacritical marks. Andre
', Mich'ele or Miche'le, and Rene'e are all examples of this, where the placement of the apostrophe seems to me to be an attempt to reproduce the French accent mark in the only way that was available at the time.
A third factor (though probably by far the least important in the African-American community) was the use by authors of many fantasy and science fiction novels of the apostrophe to make the names of their characters look more alien or exotic.
Once the idea of using apostrophes in names became established because of the above three factors, apostrophes just became part of the "respelling repertoire" for African-American parents who created new names, and so today are for many just another way to make a name truly unique. And African-Americans are by far the best cultural group at creating truly unique names. Most Americans of other ethnic ancestries will often say they want a "unique" name for their child, but when you suggest something that really is unique to them, they back off.