Same song different artist. Which ones do you love? LP

Santana She's Not There comes to mind quickly 

Two songs written  by Patti Griffins: "What you are" redone by Joan Osborne and "Goodbye" redone by Martina McBride. Both originals songs are well done, but the remakes IMHO are much better.

I probably should have mentioned "You Got What it Takes (to Take What I Got)" by Joe Tex, later covered as a duet by Marti Jones and her husband, Don Dixon. My wife and I used the latter version as our first dance.

The Feelies used to end their live shows with great cover songs - Everybody's Got Something to Hide but Me and My Monkey (Beatles), King's Lead Hat (Brian Eno), All Tomorrow's Parties (Velvet Underground). Etc.

A few stealth covers from Cyndi Lauper - "Money Changes Everything", covering The Brains, and "All Through the Night" her cover of Jules Shear.

Strange variation 1:  "World Turning", Lindsey Buckingham and Christine McVie's re-work of Peter Green's "World Keep on Turnin".  almost unrecognizable and uncredited to Green.

Strange Variation 2 -  "Battleship Chains" Georgia Satellites (Terry Anderson) covering The Woods (Terry Anderson's first band) and later covered by Hindu Love Gods (basically, REM plus Warren Zevon.)

Special Mention: Kid Creole's irresistibly danceable yet hilarious cover version of Harry Belafonte's "If You Wanna Be Happy For the Rest of Your Life (Never Make a Pretty Woman Your Wife).

Jimi Hendrix - Hey Joe

Another song w innumerable covers:
Martin Gaye's "People Get Ready" 
My favorite cover is the acapella version by The Persuasions.  Another great cover by Eva Cassidy.