To be a great cover, an artist should bring something new to the song(not just try to repeat the original performance.) You can do this by stripping a song down to its bare essence, showing it in a different light, bringing something different to the original. There were a number of great songs mentioned here (like EB noted with Nirvana) where people think the cover is the original. Let's not forget Jimi Hendrix -"All Along the Watchtower", where even Dylan acknowledges that Hendrix's version is the definitive one.
I was going to mention this song and you beat me to the punch. In fact, not only did Dylan mention that Hendrix's version is the definitive one, but Hendrix's version changed the way Dylan himself performed it. Here is what Dylan said about the Hendrix version (from wiki):
Dylan has described his reaction to hearing Hendrix's version: "It overwhelmed me, really. He had such talent, he could find things inside a song and vigorously develop them. He found things that other people wouldn't think of finding in there. He probably improved upon it by the spaces he was using. I took license with the song from his version, actually, and continue to do it to this day." In the booklet accompanying his
Biograph album, Dylan said: "I liked Jimi Hendrix's record of this and ever since he died I've been doing it that way ... Strange how when I sing it, I always feel it's a tribute to him in some kind of way."
Hendrix's recording of the song appears at number 47 on
Rolling Stone's 500 Greatest Songs of All Time, and in 2000, British magazine
Total Guitar named it
top of the list of the greatest cover versions of all time.
Studio version of Hendrix version: