There are a couple that come to mind. The first one is "The Willow Song", originally done on the Wicker Man, as performed by Brit Ekland


which was later redone by The Sneaker Pimps of their "Becoming X" album.

The second is any cover of a Bob Dylan song. I have never heard a song by him I liked but, Ironically, whenever someone has done a cover I have almost always liked it: Jimi Hendrix's "All along the watchtower" and The Byrds "Mr. Tambourine Man".

Bangles: "Hazy Shade of Winter" (over S&G)
Manfred Mann's Earth Band: "Blinded By the Light" (over Bruce)
Kirsty MacColl: "A New England" (over Billy Bragg)

Morrisseys version of Song from Under the Floorboards (Magazine)...but only just.
All of the Meat Puppets songs Nirvana covered on Unplugged in NY.

Joe Cocker The Letter over Box Tops
Joe Cocker With A Little Help over The Beatles
Ian Matthews Mercy Street over Peter Gabriel

Almost any song Ian Matthews covered over the original!

The above took 30 seconds........ I'll be back!

Hindu Love Gods (Warren zevon with 3/4 of R.E.M.)-Raspberry Beret (Prince)
Primus-Devil Went Down To Georgia (Charlie Daniels Band)
Bad Company-Young Blood (The Coasters)
Big Country-Prairie Rose (Roxy Music-and I love the original too)
Chicago-I'm A Man (Spencer Davis Group)
Judas Priest-The Green Manilishi With The Two-Pronged Crown (Fleetwood Mac)
Van Halen-Where Have All The Good Times Gone! (The Kinks)
ZZ Top-I Thank You (Sam & Dave)
The Who-Summertime Blues (Eddie Cochran)
Joe Cocker-Love Is Alive (Gary Wright)
Thin Lizzy-Rosalie (Bob Seger)
Santana-One Chain (Don't Make No Prison) (The Four Tops)
Metallica-Die Die My Darling (The Misfits)
Manfred Mann Earth Band-For You (Bruce Springsteen)

"Bette Davis Eyes" -- Kim Carnes over Jackie DeShannon
"Everytime You Go Away" -- Paul Young over Hall & Oates
"Without You" -- Nilsson over Badfinger
"24 Hours From Tulsa" -- Dusty Springfield over Gene Pitney
"Town Without Pity" -- Eddi Reader over Gene Pitney
"Downtown Train" -- Rod Stewart over Tom Waits
"Everlasting Love" -- Gloria Estefan over Carl Carlton
"Giving Up, Giving In" -- Sheena Easton over Three Degrees
"A Fairytale of New York" -- KT Tunstall over The Pogues

This was two minutes, so I may be back, too.

Manfred Mann Spirits In The Night (FIRST Version with Mick Rogers)/Springsteen
I also second For You & I'm A Man above
Any Byrds Dylan cover
Tin Machine If There Was Something/Roxy Music (Roxy's version was good, but TM's is to die for)
Commander Cody Riot In Cell Block #9 (Live version)/Any lame 50's version
Johnny Winter &/or Jimi Hendrix - Johnny B Goode/Chuck Berry
Johnny Winter Great Balls Of Fire/any 50's version

Almost any late 60's early 70's hard rock cover of almost any 50's song

Twist and Shout - The Beatles over Isley Brothers
Tainted Love - Soft Cell over Gloria Jones
On Broadway - George Benson over The Drifters
Sweets for my Sweet - The Searchers over The Drifters
Love Potion No. 9 - The Searchers over The Clovers
Ain't no Sunshine - Michael Jackson over Bill Withers
When you Walk in the Room - The Searchers over Jackie DeShannon
Apache - Jorgen Ingmann over The Shadows
I Shot the Sheriff - Eric Clapton over Bob Marley
Paint it Black - Eric Burdon and War over The Rolling Stones
I Heard it Through the Grapevine - Marvin Gaye over Gladys Knight and the Pips
Needles and Pins - The Searchers over Jackie DeShannon

Carl Carlton's version was actually a cover as well. The first hit version of "Everlasting Love" was by Robert Knight, from 1967.

Manfred Mann: "The Mighty Quinn" (over Dylan)

Lot of good ones here I might have also mentioned.

Respect - Arethra Franklin over Otis Redding ...is the only new one that comes to mind that's not already posted. I love Otis Redding, but I think Franklin's version worked for me better on that particular song.

Dylan covers have been mentioned a lot already, but:

To Make you Feel My Love - Garth Brooks over Dylan

...Dylan's version is powerful, but his voice is really hard for my ears to hear on that one.

Oh! That reminds that CCR's version of Grapevine is my fave.

Oh, crap. I need to think of these all at once... (LOL)

Both Sides Now (Clouds) - Judy Collins over Joni Mitchell
Games People Play - Mel Tillis over Joe South
You Win Again - Jerry Lee Lewis over Hank Williams (an all time fave of mine)
Nothing Compares to You - Sinead O'Connor over Prince (never heard Prince's version, so that may be cheating, but she sang that song very powerfully; hard to think his version would be as powerful or memorable)

The White Stripes's cover of "I Just Don't Know What To Do With Myself" by Burt Bacharach.
I love Roxy Music's cover version of "Jealous Guy" by John Lennon. IMHO, Ferry manages to rival Lennon on that song.
David Bowie's cover of "Hard To Be A Saint In The City" by Bruce Springsteen.

"Hard to Handle" Black Crowes over Otis Redding
"Crossroads" Lynyrd Skynyrd over Cream over Robert Johnson

Zeppelin over all their ripoffs.

I've heard both many times. Sinead's is much better.

Roxy's Like A Hurricane (also from High Road) is pretty special too; though I consider it a tie with Neil's great original. Any Roxy Fan who hasn't heard the complete High Road concert on dvd is in for a treat!

The Sugababes - "Freak Like Me" (Adina Howard)
Herman's Hermits - "I'm Into Something Good" (The Cookies)
Elvis Costello - "How Much I Lied" (Graham Parsons)
Duran Duran - "White Lines" (GF & the F5)
Soup Dragons - "I'm Free" (Stones)
PM Dawn - "Norwegian Wood" (Beatles)
Belinda Carlisle "(S)He Goes On" (Crowded House)

Dave Edmunds - Girls Talk/Elvis Costello

Without You - Nilsson
Stairway to Heaven - F Zappa
Bolero - F Zappa
the four Beatle songs Zappa covered on the 88 tour (never released on CD - yet) -I am the Walrus and the Texas Melody (Lucy in the Skies/Norwegion Wood/Strawberry Fields)
Lets Work Together - Canned Heat

Ain't That a Shame--Cheap Trick over Fats Domino
Harlem Shuffle--Rolling Stones over Bob & Earl
Working in a Coalmine--DEVO over Lee Dorsey