Where an artist played the first commercially successful version, and the songwriter later re-recorded it (i.e., Aretha's Natural Woman, written, then recorded by Carol King, who is a fine singer, but ain't Aretha), I don't really consider the first one a cover. With that in mind...
Whitney Houston's "I Will Always Love You." I like Dolly better, but Whitney did that song better.
Presidents of the United States of America: Video Killed the Radio Star.
My Chemical Romance: Desolation Row
Van Halen: You Really Got Me
Joan Jett: I Love Rock-n-Roll (this one may not count--don't know that anyone had heard of this song before she did it)
Joe Cocker: I get By With a Little Help From My Friends
Jimmy Hendrix: All Along the Watchtower