Hec already mentioned my favorite better-cover-than-the-original, Skee-Lo's "Mister Morton." Others...hmmm...
It pains me to say that anyone is ever better than Louis Prima at anything, but Los Lobos' cover of "I Wan'na Be Like You" from
The Jungle Book
swings, stomps and rocks like almost nothing else I've ever heard (eta: and that's just the studio version from the original Disney covers album - the live versions are a glorious swing-gone-wild riot). Melissa Etheridge's cover of "Maggie May" is utterly awesome. And, considerably more obscurely, Ferron's cover of "Where Is Maria?" on the Greg Brown cover album
Going Driftless
isn't quite
better
than the original, but God, it's fantastic.
Oh, I thought of one. Richard Shindell's version of "Calling the Moon" is better than Dar Williams'.
"Twist and Shout" by the Beatles.
Also? Their version of "Money".
Ike and Tina Turner, "Proud Mary." CCR's version is pretty good on its own merits (and their remake of "I Heard It Through the Grapevine" is brilliant). But only Tina thought of "We nevah. Evah! Do nuthin'! Nice! and Eeeeeeeasy."
Oh, I prefer Del McCoury's bluegrass cover of "1952 Vincent Black Lightning."
Fightin' words!
Here's a few off the top of my head:
Dinosaur Jr's "Just Like Heaven" is beloved by Robert Smith because it is awesome.
Mike Watt's "Big Train" is better than the Kinman brothers' version.
There's no better version of "The Red And The Black" than fIREHOSE's, and that includes BOC's original and the Minutemen's version.
Any song that Jerry Lee Lewis performs is by definition better than the original.
Any Daniel Johnston song performed by another artist is also better than the original, especially if that artist is Kathy McCarthy or Vic Chesnutt.
Joe Cocker's
A Little Help From My Friends
is pretty damn wonderful.
OTOH, Witney Houston's
Greatest Love of All
is a sappy mess compared to George Benson's which it went and drove off my radio...
Blue Cheer's version of
Summertime Blues.
Husker Du's version of
8 Miles High
may not be better than the original, but it's at least its equal.
Janis Joplin's "Summertime."
Whereas Hüsker Dü's version of Love Is All Around is far superior to the original.