Wilson Pickett owns “Hey Jude.” And “Silent Night.”
There have already been some ‘anything by Dylan’ votes, and I tend to agree. May I add one specific example. Tricia Yearwood’s version of ‘To Make You Feel My Love’ is infinitely preferable to Dylan’s version, and I suspect even Dylan and his mother would have to agree.
Eva Cassidy’s ‘Cathy’s Song’ is better than Paul Simon’s.
Can we have a special mention for artists who do ‘covers’ of their own work?
Sarah McLachlan’s ‘electric’ version of ‘Sweet Surrender’ is better than the acoustic version, and I think the acoustic version came first (open to correction).
I thought the Police’s (really Sting’s) second version of ‘Don’t Stand So Close To Me’ was better than the original version.
Elvis Costello’s “Good Year for the Roses” is better than the original George Jones version (though I like both).
Somewhat obscure, but I like R.E.M.'s cover of “Pale Blue Eyes” better than the Velvet Underground original. From YouTube, here they are performing it live in '84. R.E.M.'s version of “There She Goes Again” is also better than VU’s original (mainly because Lou Reed just ain’t no singer).
I love The Clash, but I must disagree. The Bobby Fuller Four kicked this song’s ass better than anybody back in '65. And, incidentally, it was a cover when they did it. The original was by Sonny Curtis and the Crickets.
Another Dylan cover: “Don’t Think Twice, It’s Alright” by Mike Ness. Kicks the original’s ass.
I should amend this: I like Bobby Fuller’s vocals better, but there’s no denying that The Clash added a whole new dimension to the song with their guitar work.
Everything But the Girl’s version of Elvis Costello’s “Allison” (their version of Lauper’s “Time After Time” is great, too)
Gomez’s version of The Beatles’ “Getting Better”
Prince’s version of Sheryl Crow’s “Every Day is a Winding Road” (with Crow on guitar, I believe)
Concrete Blonde’s version of Leonard Cohen’s “Everybody Knows”
Frente!'s version of New Order’s “Bizarre Love Triangle”
Lilly Allen’s version of Keane’s “Everybody’s Changing”
Sinead O’Connor’s version of Prince’s “Nothing Compares 2 U”
I’m sure I’ll get flamed for this, but I think Aaron Lewis’ cover of ‘Black’ was far better than Pearl Jam’s original version. Not to say that the original wasn’t awesome, as well.
I’m sorry, but if you insist on claiming that Whitney Houston’s version of that song is better than Dolly Parton’s, I may have to demand satisfaction on the field of honor.
Of course, having said that, I must admit with some shame that I really used to like the Bangles’ version of “Hazy Shade of Winter.”
Of course, for truly fine covers you can’t beat Pat Boone singing “Smoke on the Water,” “Panama,” and other metal hits.
Yes, we are talking better here, not more famous (My Boss once asked me to print out the lyrics to Whitney Houston’s “I Will Always Love You.” She gave me hell for printing out Dolly Parton’s.)
More Dylan with Baez’s “Forever Young” and “Lily Rosemary & The Jack of Hearts.” Joannie covers Bob better than anyone else.
Diamond David Lee Roth (and the rest of VH) put a lot of great energy into superior covers of tired originals.
California Girls
Pretty Woman
Just a Gigolo
Where Have all the Good Times Gone
Dancing in the Streets
Ice Cream Man
You Really Got Me
I really love the Rolling Stones’ version of “Around and Around” - one of the best Chuck Berry covers ever. Especially cool is the version from the TAMI show, when Chuck was in the building.
Sadly, I don’t like any of the Beatles’ Chuck Berry covers. Paul could pull off Little Richard, but “Roll Over Beethoven” and “Rock and Roll Music” seem a bit pale compared to the originals - and poor when stacked up to what contemporaries like the Stones and Yardbirds were doing.
Dang-nub-it! Beat me to it, I came in here to post:
Sympathy (Rolling Stones) - Jane’s Addiction
Like a Rolling Stone (Bobby Dylan) - Black 47
Planet Earth (Duran Duran) - Hate Dept.
How Does It Feel? (Blue Monday / New Order) - Electroset
Ooh! I love that song. I’ve got the original record somewhere on a cassette tape. The UVic library had it, for some reason.
Clive Gregson and Christine Collister’s version of Jackson Browne’s “For a Dancer” is my contribution.
I’ll have to find that. I’ve always thought that Susan Tedeschi’s verion of "Don’t think Twice It’s Alright " was fantastic.
The Monkees’ version of “Daydream Believer” was better than Neil Diamonds’ and Smash Mouth.
Live’s version of Johnny Cash’s standard “I Walk the Line” is better, IMO.
Sarah McLaughlin’s version of Joni Mitchell’s “River” is awesome.
Eva Cassidy’s “Fields of Gold” is equal to Sting’s original.
Luther Vandross’ “All the (Wo)Man I Need” is more bluesy than Whitney’s.
Oh, and Dolly Parton’s version of “I Will Always Love You” is still the best version, IMO. It’s supposed to be a humble song, for pete’s sake.
Sinead O’Connor’s version of Bobby Gentry’s “Ode to Billy Joe” is incredible.
Finally, there’s Janis Joplin’s wonderful version of Kris Kristofferson’s “Me and Bobby McGee”
The Firebirds cover of Tom Paxton’s “Bottle of Wine” was a hard rocking hit!
Just to be different:
The snippet of Fell In Love With a Girl that Weird Al Yankovic includes in Angry White Boy Polka is sung vastly better than the Whitestripes version. 
Now I gotta be spending the rest of my day over at Second Hand Songs
I am a little surprised you did not mention Pat Boone’s cover of “Tutti Frutti.”
Every single song on Ray Charles’ “Modern Sounds In Country and Western Music Volume One” is better than the original.
Ditto for Emmylou Harris (even her Chuck Berry cover - C’est la vie (You never can tell) is as good as the original.) Her version of “Sweet Dreams” e.g. has it all over Patsy Cline.