Gender bender

Cowboy Junkies (everything)
--I was thinking of their version of Neil Young’s Powerderfinger, or One by U2, but you have a point. (both youtube)

Can’t Find My Way Home - Rachael Price (youtube)
--Hard to beat Blind Faith, but a great cover.

Jennifer Warnes--Famous Blue Raincoat (Leonard Cohen)
--Between the sax and her voice...

Bow Wow Wow’s version of I Want Candy comes to mind, originally done by The Strangeloves. Whether or not it’s better than the original - ?

+1 on the Sundays cover of Wild Horses. 

Also, Macy Gray's cover of Metallica's Nothing Else Matters is pretty good. 

Leonard’s Cohen’s "Alexandra Leaving" done solo by his long-time collaborator Sharon Robinson

This one is a close call but Holly Cole’s "I Want You" is about dead even with Tom Waits’ original.

Not sure if any would agree is truly "better" but certainly a whole different spin and genre change here.

Judas Priest cover of Joan Baez Diamonds and Rust.
Appeared on their Sin after Sin album.
Personally I love what they did with it.