First off, he'd have to start a new twitter and who knows how many followers he would lose.
Second, a lot would depend on if he used the name. Smaller indies will use it anyway, but the bigger guys have to be more careful.
I think a guy like Barretta will do GREAT, because he can really go in the ring, so he will get booked by places like PWS, NWS, etc that like to use former names, but also the 'workrate' indies that like to use great workers and put together great matches. His phone literally rang as soon as he was fired, and he was booked by PWS before WWE even announced he was gone. For a guy who was never used on house shows, didn't work a single PPV, had no merch, wasn't on many (if any) DVD's, wasn't in the video game, etc he was basically making his downside and might do just as well (if not better) on the indie scene (at least the first year or so).
But what is Ryder worth without the name? I don't know. LIke most he's do well for a few months for sure, but beyond that, the further he gets from TV it becomes a struggle.