Your question assumes a basic circumstance that is untrue: that captive Africans and their descendants had the power to name themselves. While some African names persist throughout African chattel slavery in America, they are not nearly as common as European and Ancient Greek/Roman names. Furthermore, the names that enslaved Africans were known by depended largely on the culture of the people who owned them. So while Jim and Tom may be common names for enslaved people in the Upper South, Francophone names like Celestin, Jean-Baptiste, Francoise, and Marie-Joseph are very common where I work in South Louisiana. Spanish names also appear, owing to the Spanish control of the territory. However, classical names are extremely common throughout slavery - names like Plato and Aristotle, e.g. A mentor of mine explained this to me, sadly, as an expression of a cruel joke. Enslaved people were also named after characters from pop culture. Some names from plays and literature appear over and over again. Generally, traders and owners gave enslaved people names they were familiar with. There is also no doubt some correlation between giving captives European names and the impulse to "civilize" them, particularly with regard to religion (esp. in Catholic territories).

Generally, enslaved people had a tenuous grasp on their names. A very common aspect of chattel slavery that comes up over and over again in narratives is the loss of one's name, or being re-named. Just one very famous example of this is that of Solomon Northup who wrote 12 Years a Slave. During the entirety of his bondage in Louisiana, he was enslaved under the name of "Platt". He wasn't the only person - not by a long shot - who had a slave trader give him a new name. "Lost Friend" notices in newspapers after the Civil War show us this as well. When looking for family members they lost in slavery, freemen did not always know their names. They'll say things like "My mother's name was X but the speculators called her Z"

Curiously, naming traditions after slavery show us that people didn't really forget, generations on, who they were. While European names dominate early freed societies, the last names they took are revealing in some circumstances. Generally - not always - enslaved people were only known by first names. After the war, they have last names and historians are left to puzzle at how they were chosen. Some took the names of their owners but not all. In South Louisiana, some common last names reveal African identities -- Sinegal, Ibo, and Poullard are all last names that appear in the postslavery period, which tells us that even decades after the end of the transatlantic slave trade, people had names pass down through their families.