I was just talking about a remake for Red/Blue today with my girlfriend... I would want a whole new story.
Like a whole new generation of trainers and Gym Leaders and a TON of references to Pokemon Red/Blue original or FireRed/LeafGreen.
I don't just want a remake...

It'd be cool to have all the newer features (including the cool battle types, and more of them).
An original story (for once) would be great too.
Like ALL the Gym Leaders from the old game have passed on the title to a new leader... As if 15+ years had actually passed between then and now. It'd bring the whole series full-circle and introduce a whole new generation to the original Pokemon.
I wouldn't mind seeing "newly discovered" Kanto-Pokemon either, it'd give Gamefreak/Nintendo another reason to remake them...
I want to feel like time has passed since I last showed up and that their society has changed (reflecting from the battles, gym leaders, playable characters and so-on).

Only thinkg I DON'T want is a "fully 3D" game that someone mentioned above... I prefer the 2.5D (2D-asymmetric view) of the Pokemon series... It's a signature of the series... And fully 3D games still don't look the best on the 3DS (though MUCH better than on the DS/DSi).