Slavery existed in the southern states of the US until after the Civil War. This terrible system of bondage brought thousands of Africans to the Americas against their will. When slavery was abolished, the former slaves needed to decide what family name they wished to use. Often they took the name of the family for whom they had previously worked. However there was no set rule.
There are a number of families of African descent with the family name Bitting. The second federal census following abolition in 1880 lists 11 heads of families and 63 individuals of African descent in North Carolina and Georgia. A listing of returns from the 1880 census in which at least one person in the household had the name Bitting and who gave their race as Black or Mulatto can be found here. At least one of the families listed in the 1880 census, Elias Bitting of Winston, NC, still has decedents living in North Carolina. Some of these people lived in an area of Winston-Salem called "Happy Hill".
There are many on-line resources for anyone interested in doing more research on this subject. A good starting place is Cyndi's List for African-American resources.