I kinda slammed promiseoftin for her answers to this one, so I feel almost compelled to answer and submit myself to the same potential criticism.
My immediate thoughts were The Bangles' cover of Simon & Garfunkel's "Hazy Shade of Winter" and Siouxsie and the Banshees' cover of The Beatles' "Dear Prudence".
For the longest time -- as a child of the 80s -- I never knew the S&G version of "Hazy Shade of Winter". I had to discover that later on and that puts me in mind of another great cover. Jimi Hendrix's cover of Bob Dylan's "All Along the Watchtower".
The Dylan version just doesn't come close.
One of the biggest complaints I have about covers -- and The Ataris' version of Don Henley's "Boys of Summer" is a great example -- is that most of them aren't re-imaginings. They're covers in the original sense of the word -- exact copies of the songs that you'd get someone to perform. Hearing The Ataris' one is almost exactly like listening to the original, so what was the point of going through the effort of recording it? To prove that you can sound like Don Henley?
There's an amusing cover competition to see who can perform R.E.M.'s "It's the End of the World as We Know It (and I Feel Fine)" in the shortest period of time. It's never going to produce a better version, but it does produce entertaining ones.
I guess there's something to that.