Gender bender

Name a cover that is better than the original sung by the opposite gender.

Angel by Lucinda Williams and Charles Marvel. (Jimi Hendrix)
Wall of Death by Shawn Colvin. (Richard Thompson)
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.