When it comes to naming slaves, we have limited data to work from. The vast majority of slaves born in the United States were only recorded as a number in the decennial censuses. But from the data we have available, from runaway advertisements, inventories, diaries, wills, and letters, the evidence seems to indicate that slaves named their children instead of the slave owner. There is a fair amount of literature on slave naming conventions, and I want to point to a few I'm drawing data from: Phil Morgan, Slave Counterpoint; Cheryl Cody, "There Was No 'Absalom' on the Ball Plantations" American Historical Review, 1987; Burnard, "Slave Naming Patterns" Journal of Interdisciplinary History, 2001; and Handler & Jacoby, "Slave Names and Naming in Barbados", William and Mary Quarterly, 1996. I am also drawing on my own research into the Butler family of Maryland, a group of freed slaves in the early national period.
In her essay, Cody explains that when a slave purchaser or master buys a slave, they try to use distinct names or descriptors in order to make sure that the slave is uniquely identified. This makes it easier to assign slaves tasks, divide them in probate inventories, and to identify them in runaway advertisements. The slaves that the masters did name were typically ones imported into the colonies (or later United States) from Africa and the Caribbean. So in runaway advertisements for recently imported slaves, we are more come across names like Caesar, Scipio, Cato, and Scotland. In inventories or wills, when trying to distinguish between two slaves of the same name, the master would use descriptors like "Big Pompey" and "Little Pompey" (Cody, p. 572).
Counter to this preference among slave masters was slave preference for naming children after other family members. Among slaves owned by the Ball family, one son in six was named after their father; a bit less than one daughter in ten was named for their mother. More often slaves chose to name their children among wider kinship networks including grandparents, cousins, uncles and aunts. Among the Butler family in Maryland names like Ignatius (and derivatives Nace, Nat, Nate) Elizabeth, William, and Rachel predominate. Some slaves chose to name their children with African names to tie the children to their homeland. names like Cudjo, Cuffee, Sambo, Amboo, or Juba. Slaves also had the ability to fashion surnames for themselves. Typically slave masters did not give their slaves surnames, but admitted that the slave had made a name for themselves. So Dick became Richard Wallace, Isaac became Isaac Elsa, Billy became Will Short, and Mareen became Mareen Boston. Slaves tended to avoid using the surname of their master. Some slaves did, but they were typically personal servants like Henry Lauren's slave Robert Laurens. Most instead chose names of their own, or of names of free black families. Butler, Boston, and short are examples of this.
None of the above patterns seem to fit the descriptions of how slave masters would name slaves. It is a contentious topic because the ability for a slave to name their child is a basic component of building a family. There used to be more of an argument over the destructiveness of slavery - was slavery so bad that it destroyed the slaves capability to maintain a family and familial relationships? Research and writings of the past half century have (fairly definitively) proved that as damaging and horrible as slavery was, the enslaved Africans and African Americans were determined to make lives for themselves. They married (even if the marriage meant nothing in the eyes of the master or the state), they named their children after family members, they trained them in whatever the parents knew, from "plantation work" to smithing, cooping, and boating. Slaves absented themselves from plantations to visit family that were either owned by other masters or had been sold away. Slaves were absolutely able to build families and relationships outside the plantation or farms boundaries, partly in defiance of their masters (and partly with their tacit consent).
A slave master had the greatest power over a recently arrived slave from Africa. Most often they spoke no English, had suffered brutal hardships on the African coast and in the belly of slave ships. The slaves tried to build relationships with "ship mates" as best they could. The Africans had names from their homeland, but in an atmosphere where assimilation into the existing slave population was made a priority by masters and overseers, enslaved Africans often had to take on a name assigned them by the slave purchaser. Some slaves tried to keep their African name, (Morgan p. 455) so that Bristol went by Cuffee, London to Appee, and Charlestown as Tamoo. Part of the reason I mentioned in the beginning that the majority of slaves were recorded as just a number is to help point out that we only now a fraction of what was happening on the ground among slaves. There are a few examples of Africans trying to keep a past name and life, but far greater is the preponderance of silence in the sources. African names died out over the course of the eighteenth century in the Chesapeake, as more slaves adopted forms of Christianity or other aspects of white culture. That did not stop them from asserting their own individuality and building families, but most of what we see of this resilience is through the hostile eyes of overseers and plantation owners.