Besides some of the ones listed (Jimi Hendrix's "All Along the Watchtower," the Animals' "House of the Rising Sun"), I also like:
-The House version of "Get Happy." They put it in a minor key, and it sounds frickin' awesome. ...Except when it gets all happy/silly, so I skip that part.
-Peter Gabriel's version of Arcade Fire's "My Body Is a Cage."
-PHEVOTT and Tsubaki Ichimatsu's version of ZUN's "Kid's Festival ~ Innocent Treasures." The original's interesting, but I just like the other more.
-Masayoshi Minoshima and Nomico's version of ZUN's "Bad Apple!!" The nice black and white video doesn't hurt. :D
-Blackmore's Night's version of Renaissance's "Ocean Gypsy."
-Loreena McKennitt's version of "The Highwayman." It was originally a poem, and the only other musical version I've heard (Phil Ochs, I think) I didn't care for at all.
-The Pokémon Live! version (as opposed to the third movie CD version) of "You and Me and Pokémon."
-The show version (as opposed to the 2.B.A. Master version) of "The Time Has Come (Pikachu's Goodbye)."
-I'm not actually sure which came first, and they're both by the same people , but I like the Engrish "Just Like a Rainbow" over its Japanese counterpart "Niji ga Umareta Hi." Listening to "Niji" again, the timing of the words is just off as hell. Sometimes (mostly at the beginning) it sounds like she's trying to squish too many syllables into too short a time, and other times it sound like she started a word too early and just kept singing it until it should've ended.