I am going to put my hefty disdain for the JJ film aside, and actually try to offer a serious answer to the OP's question...
As much as I did not like the way things were done in the film, and the direction JJ went with it, I have to admit, I did enjoy it, as brainless fun. But there were a few things I would have done, to make it work a little better...
Number one, get rid of the ice monster scene. It didn't really contribute anything to the movie, and the monster was kinda lame anyway.
Two, show more of the background with Nero... include the stuff that was cut, dealing with Rura Penthe, or show his attack on the Klingon armada. I left the movie wishing we had spent more time with Nero.
Kirk should have had a damn yellow uniform throughout the film. I am SO sick and tired of the whole "oh, he's the bad-boy rogue type, who rebels by thinking for himself, so because of that, he should never wear a standard uniform" mentality. Enough with it already... he was in Starfleet, give him an appropriate uniform, and not that all-black piece of shit he was wearing for half the film.
Redo the Enterprise's Engineering section. Just redo it.
Finally, this is something that a lot of people might not agree with, but I still think would have been AWESOME...
The more I look at the film, and it's flashy new look, and use of lens flares, the more and more I am absolutely convinced, that this film would have looked 110 times better, if it were filmed in black and white. Bear with me now... I TRULY think that if you were to watch the film with the color removed, it would look not only totally awesome, but would be a cool nod to the TOS pilot, as well as be a refreshingly artistic move, on the part of the director... a brand-new, flashy, modern sci-fi film, done in black and white... I REALLY think this could have worked well, and I challenge anyone to actually watch it in black and white, and tell me I'm wrong... it would have looked refreshingly artistic, and very TOS-like, yet new.