Cover Songs that are better than the Original

Eva Cassidy - Fields of Gold
Eva Cassidy - Over the Rainbow
Eva Cassidy - True Colors
Eva Cassidy - Time after Time
Eva Cassidy - Autumn Leaves
Johnny Cash - Hurt
The Who- Rush cover - The Seeker
Rush cover -The Pass  Fleesh Version

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