Uncle Jeff is his nickname. It's a done deal; he is unavailable for nicknaming. New nicknames never stick, and they are usually just annoying. Case in point, anything Shaq ever tried to call himself, anything Big Baby ever tried to call himself. The other thing, though, is that it actually fits. Jeff Green is a quiet, stoic type, a solid dude in the background that is always there. The uncle thing works.
It is also too late to rename Steez, Jet, the Truth, Jesus Shuttlesworth/Ray-Ray, the Big Ticket (ugh... worst nickname ever), Big Baby, Perk, Nate the Great, Starbury, Hump, No-Pass Bass, or Crash.
You could try with Vitor, but I don't think anyone's coming up with anything better than El Hombre Indestructible. That's on the fast track to being one of the coolest nicknames of all time.
If you want to nickname Sullinger, you might be able to get in there before "Sully" is cemented. It seems inevitable, though (sorry, Smokey the Swag Bear). Likewise, KO seems inevitable (what about Sunshine you guys?! It's totally a reference to sports movie and not racist in any way!).
Angry Bird never really caught on, and pit bull never will either... so Avery Bradley is still available.
Courtney Lee has no nickname, but that's because he never did anything you wanted to nickname him for.
Rajon Rondo has no nickname, which is kind of amazing. People have tried for years, and they all seem to suck.