I'm surprised one of history's most famous cover songs, I Love Rock and Roll by Joan Jett & the Blackhearts wasn't mentioned yet. It was originally performed by the Arrows. Even most reasonably music savvy people probably don't even know who the Arrows were. I know I didn't until I was told the song wasn't originally sung by Joan Jett and looked up the info. Obviously her version is much better.
Another one that comes to mind is Quiet Riot's Cum on Feel the Noize which is a cover of the song made by the band Slade.