I have a few ideas of reboots I'd like to see.
Omnikron; I spent countless hours on this game when it first came out. It had some interesting features (though the shooting segments sucked so much), and an interesting story (player is sucked into the universe as a digital ghost that can body jump, and depending on who you jumped into, it could alter the story).
Messiah; Another game I spent a lot of time with. In this game you play as a cherub or Putto (think baby angel with a bow), that was sent to earth by god to clean up mankind as they have become corrupt. God eventually decides that mankind is too corrupt and that they want to mess with his design, so he tells the angel to return to heaven, and he declines. Then Satan starts to follow you around and tries to sway your actions. The main fun of the game was possessing different people and causing havoc. The puzzles were fun and unique, and the gameplay itself was very interesting.
Shenmue; There has been some talk about making a sequel recently, but I'd much prefer them to just reboot it at this point. Right now the original game is so old, that a good portion of the target audience wouldn't even recognize it. Still though, this was probably one of the first open world free roaming style games that managed to pull off a semi-realistic day to day feel, while still being very fun.
Parasite Eve; I don't want a true to form reboot, in fact I'd like to see it change the style of game play from turn based rpg, to something more like Skyrim or Fallout, but keep the horror aspect and general story concept the same. I would even be okay with it being set up like RE4/5 or Dead Space 1, however they would really need to play up the horror aspect, and we would need someone who can actually design a good horror game to be in charge of it, otherwise it would be better off sticking with an RPG style game (just real time and not turn based).
Earthworm Jim or Vectorman; I'd like to see these games rebooted as a 3d adventure/platformer game, and it wouldn't be too difficult to set up (just look at Prince of Persia, but different movement mechanics). These games were typical platformers of the 90s, but they both separated themselves by having fun mechanics that really made them stand above the others of their time.
Conker's Bad Fur Day; This was probably the single funniest game I've ever played. At one point in the game, you literally have a boss fight with a giant opera singing turd. If you haven't ever played this game, find someone with a N64 and play it, I mean right now.
Jedi Knight Series; With all the attention on Star Wars right now, this would be the perfect time to bring back a Star Wars game that didn't focus on the Jedi/Sith conflicts. With the new technology out there, it could be so much better than it was back when it first came out, and the basic design principles are already in place.
Shadows of the Empire; Since I am already on Star Wars, this is probably one of the best Star Wars games ever released. I still play this game from time to time, and enjoy it. However, Disney would have to choose between this or my previous suggestion if they wanted to do a reboot. There is no reason for both to be produced at the same time, but if one is successful, they may consider the other.
Ultima series; I'd like to see this rebooted and made into a 3d, 3rd person, action RPG set up similar to Dark Souls, but with less focus on fair but brutal difficulty, and more focus on solid combat with good story telling.
I could probably go on forever with this, so I'll just list the last few on my mind and move on. MediEvil, Freespace, Vagrant Story, Wild Arms, Bushido Blade, Syphon Filter, and Revolution X.