A couple spring to mind immediately, and I'm sure more will come:
Sid Meier's Covert Action: If this could get the same sort of treatment as Pirates! did a few years ago - simplified a tiny bit but mostly just brought up to date - it would be all kinds of win.
Killed Until Dead: This C64 game was kind of like real-time Cluedo, where you had to keep an eye on who was doing what during the evening, rather than just finding the facts afterwards. The concept was really clever and I haven't seen it since. I think I'm more hoping for a new game with the same concept, than a remake of the original game.
P.S. Great idea for a thread!

Perfect Dark!
oh wait, that's getting a re-release with updated textures and xbox live functionality..
TMNT: Turtles In Time!
oh wait, that's getting a remake in full 3d and xbox live and psn functionality...
Daggerfall, but it has to be only a graphics update.

Don't even think about it. It's also going to have very gay facial animations and only 6 skills to choose from

Daggerfall: Updated and more bugfree, but still the sprawling game it was, anything else and a pass. Mod-able also a bonus.
Insert random RPG here.
A collection of all the old c64 games would also be nice.
Eh, not much else really. Old favourites still have their charm, and modern remakes have a habbit of catering to mass market appeal when the original games were more niche.

Heroes of Might and Magic 3. Just the same game with better graphics, a few more units, one or two new campaigns, and the same addictive hotseat mode.

I never played the first one. Sounds like I should, if I can get it running.

Sounds like you never played Murder on the Amiga.

If the concept was the same, then that deserves a remake too :)

Ecstatica and Little Big Adventure.

Dang... don't make any quick movements gramps, your pacemaker can't handle it. :D
Kidding, if it helps I'm right there with you on that. I remember when those games were the new hotness.

Why? They're not that old. I love Little Big Adventure (Twinsen's Odyssey), too bad I never got very far with that. But it was fun, and somewhat revolutionary for its time. Ecstatica, I never got past the first level with that lol, simply because I didn't know how to handle the controls, but it had a decent story if I remember

Dunno, it just seems like an absolute age ago when I remember those coming out.

Carmageddon 2. Not censored.

Remakes are a bad idea. They tend to make bizarre changes to level design or art direction, remove features the original game had, or otherwise harm what made the original so good. Of course, none of this applies to "pure" remakes (like the HD titles on Xbox Live Arcade) that just put a fresh coat of paint over the original gameplay.

I'd go for Banshee on the Amiga. Though I have no idea how you'd change it from top-down. Maybe just have it in HD would be cool