Surely this has to be the Gold standard.
[Jeff Buckley - Hallelujah]
Even the fact that every hipster douchebag plays it at parties to impress girls doesn't diminish the sheer raw beauty of Jeff Buckley's version, which may well be his masterpiece.
I was gonna say this, though really most anything Jeff Buckley covered was better than the original. His covers of "If You See Her, Say Hello" and "The Way Young Lovers Do" on the Live at Sin-e album are awesome.
Most of that album is simply heavenly. Rest in peace Jeff Buckley. I cant think of many covers I like more than the originals but I do enjoy the occasional cover from time to time. The first thing that comes to mind is the cover of Hey Ya by The Blanks.