I'm on the bandwagon for Sonic 06'
These would be a few things I think would make a re-release of this game amazing
Sonic: Normal Stage gameplay would play out alot like Adventure 2. Clearly by now Sonic Team could make that formula work without bad camera/slippery controls by now...right? For the Mach Speed sections, have it shift to Unleashed/Colors/Generations (Modern) Gameplay.
Shadow: I never did unlock him (I've raged quit Sonic 06' more times than any other game)
Silver: Speed him up, make his controls tighter, make his levels and abilities more varied and more interesting.
Side Characters: I only had the chance to play with Blaze and Tails, not sure how the others played, Tails was aweful, if they make him playable, remove the fake ring bomb thing and make tighten up the way he flies. Blaze was actually alot of fun to play as, I think in general if they just tightened the controls it would make it more enjoyable.
Story: Finish it...remove the weird Elise/Sonic thing, no one will mind if you update/change the story.
Music: The best part of the previous version of the game, leave it as is.
A few extra things I can think of
-either shrink the hub world, or make the old hub world more interesting an varied. Sonic .
Unleashed did a fairly good job with it's hubworld, and I think they could pull off something similar for a Sonic 06' remake
-although not originally in Sonic 06' I think a Chao Garden would be a extremely welcome edition to the game. If the remake was made for the Wii U, there would be very intesting things you could do with the 2nd Screen, or possibly a mini chao garden app on the 3DS...