Cover Songs that are better than the Original

Jimi Hendrix - All Along the Watchtower
Bonnie Raitt - Angel from Montgomery

I once heard a version of Hallelujah in a small pub in the middle of Wisconsin that absolutely blows away every version I have ever heard. It was performed by a singer/songwriter from Minnesota and other than his guitar and vocals, you could have heard a pin drop. The whole place was mesmerized.

Hallelujah - Jeff Buckley
Suzanne - Judy Collins

Hendrix-All Along the Watchtower
Righteous Brothers-- Unchained Melody
Roy Orbison-- Love Hurts 
Willie Nelson and Daughter--Have You Ever Seen the Rain
Yoko Ono--Someone Like You

Not necessarily better than the originals, but, I like these covers:

Concrete Blond--Everybody Knows
Odetta--Times they are a Changin