Speaking of "Sweet Jane" by The Velvet Underground, how about the cover of that same tune by the Cowboy Junkies? I doubt that most Cowboy Junkies fans know the provenance of this tune.
Going one step beyond "cover songs" and into the realm of "cover albums" you gotta love George Benson's cover of The Beatles' "Abbey Road." That was back in Benson's jazz days. About two years later, in his album "White Rabbit," Benson did some great covers of "White Rabbit" (originally by Grace Slick and Jefferson Airplane) and "California Dreamin'" (originally by John and Michele Phillips of The Mamas and the Papas).
Going one step beyond "cover songs" and into the realm of "cover albums" you gotta love George Benson's cover of The Beatles' "Abbey Road." That was back in Benson's jazz days. About two years later, in his album "White Rabbit," Benson did some great covers of "White Rabbit" (originally by Grace Slick and Jefferson Airplane) and "California Dreamin'" (originally by John and Michele Phillips of The Mamas and the Papas).