This doesn't have to be a loaded question...but why not let loose and give an ear-full...or and eye-full...or whatever voice you hear in your head when you read full.
What i mean by remake is: keep the storyline and characters of that specific game and installment (which game in the series such as Mario "2"), keep the same core elements and identifying characteristics of game play, keep the same major musical pieces, just apply those things to the new medium and expand on those things immensely. A good example would be the Metal Gear Series.
1. Splatterhouse (turbo Grafix 16)
2. Castlevania 2 (NES)
3. E.V.O. (SNES)