How were surnames attached to deceased slaves, prior to 1862?

I ask this question because I have about 70 African American family gravesites of ancestors and relatives, scattered throughout Nelson County, VA - extending from Albemarle (near Charlottesville), down to Lynchburg.
I am able to read the hand-carved names of relatives carved into large stones, who died in the 1870's.

But I've also noticed names of relatives carved into these stones, or rocks showing their lastnames, who died in the early 1830's.
Since the slave owners of this area noted in their diaries, that they or their farm mangers, and overseers did not assign thier own last names to their slaves, is it possible that the children.. now freed slaves after 1865, went back to their parents burial grounds, and hand-carved the names, along with their newly assigned surnames, into these old stones, and attached their name to their parents, in order to give them an identity as a memorial, and rememberance?