Basically any cover Johnny Cash did qualifies most of the time.
Mouth of the Architect's cover of "In Your Eyes" (original by Peter Gabriel). I mostly like it because I never thought the original was particularly outstanding (it wasn't that bad, though), and the MotA cover is...well, I shit myself with joy when I hear it. It also is basically a different song, save for the lyrics.
Cee Lo Green's cover of "No One's Gonna Love You" (original by Band of Horses).
A Perfect Circle's cover of "Imagine" (original by John Lennon) is a bit better than the original, I think. Also most stuff on the eMOTIVE album. "Annihilation" (original by Crucifix) is hauntingly beautiful, where the original was kinda...I didn't like it. "Gimme Gimme Gimme" (original by Black Flag), more or less same effect. Original was all right, cover was incredible. And so on, by and large.