I don't know if better is the word I'm looking for, but one song does stick in my mind. John Lennon's "Imagine" was covered by A Perfect Circle, who did it in a minor key. It turned the song from a plea for love and peace into something fascinatingly dark and mysterious (though still about love).
One that I thought was definitely covered better was Cyndi Lauper's "Time After Time," which was covered by Quietdrive. Quietdrive's version is actually something I could listen too over and over again, while I have a limit on how much Cyndi I can take in a session.