There's a lot out there, top of my head:
-Dylan's "One Too Many Mornings" covered by Jerry Jeff Walker on A Man Must Carry On
-Dylan's "If Tomorrow Wasn't Such a Long Time" covered by Kingston Trio
-Paul Simon's "Bridge Over Troubled Water" covered by Elvis Presley
-Kevin Farrell's "Sonora's Death Row" covered by Robert Earl Keen
-Burt Bacharach's "Walk on By" covered by Diana Krall
-Prince's "Nothing Compares 2U" covered by Sinead O'Connor
-Henry Glover "California Sun" covered by the Ramones
-David Bowie "The Man Who Sold the World" covered by Nirvana
-JJ Cale "Cocaine" covered by Eric Clapton
-Top Notes "Twist and Shout" covered by The Beatles
-The Beatles "Get by With a Little Help From My Friends" Joe Cocker
Up for debate?:
Black Magic Woman - Fleetwood Mac (original) vs Santana
Me and Bobby McGee - Roger Miller (original) vs ...everyone