I don't know what to call this, but I'll use Castlevania as an example. You know how Super Castlevania IV has newer renditions of songs from the NES Castlevania titles? Well, which other games have that?
To start off:
Phantasy Star IV: Ladea Tower's music sounds like the dungeon theme from Phantasy Star 1.
Super Mario Bros. and Zelda have too many to count.
Metroid Prime has a new rendition of the Brinstar theme in the beginning.