I love the Carolina Chocolate Drops' cover of "Hit 'Em Up Style," which in the original is the sort of problematic break-up song that makes you go, "Why is this tune set to these words?" It gets the full revenge-driven angry person treatment in the cover.
Elliott Smith can cover many songs well.
I think people are surprised by how many Beatles songs were actually covers because they did such a decent job: http://en.wikipedia.org/wiki/List_of_songs_covered_by_the_Beatles
. I'll admit that Ray Charles did "I've Got a Woman" better; Chuck Berry's "Rock & Roll Music" is better, too; there are others, e.g. "You've Really Got a Hold on Me," which is covered a lot. But many times it's either even or the Beatles did it better.
One of my favorite musicals, "Tommy," is funny because The Who recorded a whole album of their versions of their songs, which I love, but Elton John doing "Pinball Wizard" is so great; Tina Turner and Eric Clapton are fab, too, but they probably could've skipped Jack Nicholson singing.