The records I have transcribed so far include:
Owner Emancipated slave
William Cain Africa Parker
Ave Hathcook Jacob Johnson
John Hogg Frank Hogg
Pomfrit Herndon Peter Loftus
Richard Herndon Daniel Herndon
Pleasant Henderson Abraham Mannis
And I have many others to transcribe, but it appears from the transcripts (at least in Orange County) that the slave has a choice in the name chosen. I can't guess why your family ended up with the name of Green, I don't know the relationship they may have had with their former slaveowner, I would imagine that paternity was not the only determination of the assumed name. I have many ancestors who assumed surnames that differed from that of either of their parents and I can't even begin to guess why from this amount of time and distance! Your guess is as good as mine!
Gwen