Well, the first one that came to my mind was Joe Cocker's titanic "With a Little Help from My Friends," largely because I'm in a Woodstock frame of mind. Cocker also managed to make an actual full-length song out of the ABBEY ROAD medley track "She Came in Through the Bathroom Window"
Siouxsie and the Banshees covered "Helter Skelter" on THE SCREAM, and did it in a way that made it fit right in with their own songs. (Their version is certainly a hell of a lot better than the one on RATTLE AND HUM.)
I've referenced this before on the forum, but my favorite Beatles cover is a comedy record: Peter Sellers's DR. STRANGELOVE-flavored version of "She Loves You", produced by none other than George Martin himself. Sellers recasts the song as a conversation between his Strangelove persona and a lovelorn Nazi officer, with hilariously sinister interjections:
"She said you HURT her so..."
"Goot!"
"She almost lost her mind..."
"Goot! Ist goot!"
"But now she says she knows zat you're not the HURTING kind."
"I AM!!!"
"She says she LUFFS you!"
"Ya?"
"Ya. Ya. She LUFFS you."
"YA?"
"Ya. Ya."
There were plenty on "Sam I am Sam" soundtrack and some were pretty good.
I especially liked Revolution by Grandaddy, strangely bittersweet.
And as I can not cross any topic of this forum without talking about Elliott Smith, he has made a great cover of Because on the soundtrack of American Beauty.