These came to mind:
Jenny Lewis cover of "Handle With Care"
Annie Lennox cover "Georgia on My Mind"
Kelly Hogan cover "Daddy's Little Girl" (M. Ward)
Cowboy Junkies (everything)
The Sundays - Wild Horses
Aretha’s cover of “Respect” (Otis Redding)
Joan Osborne's version of Slim Harpo's "Shake Your Hips." Smokin' hot! Her version of "What Becomes Of The Broken Hearted" is also fantastic, though maybe not "better" than the original by Jimmy Ruffin.
From the opposite end, in 1956 Elvis leveraged his popularity to sell 10 million copies of "Hound Dog" which was originally performed by Big Mama Thornton in 1952. Big Mama's version was infinitely better and helped shape rock and roll for guys like Elvis who profited to an incredible degree from her work on this particular cut.
Not sure about better, because I love both versions - Rickie Lee Jones, Show Biz Kids (Steely Dan)
Buck Owens' Together Again performed by Emmylou Harris. Okay, nearly everything cross-gender covered by Emmylou Harris.
Kate Bush's "Rocket Man."
John Hart's Jazz Guitarist instrumental version of Joni's Both Sides Now. Fantastic IMO!
Joan Baez doing Stevie Wonders ,I never thought you'd leave in Summer.
Judas Priest, Diamonds and Rust (Joan Baez)
Amy Helm cover of "Mandolin Wind"
Rumer (Sarah Joyce) doing Paul Simon's Long Long Day
Lowell George doing Rickie Lee Jones' Easy Money
The Bangles girl version of Rundgren's Open My Eyes
Janis Joplin “Me and Bobby McGee” (Roger Miller)
Joan Jett’s cover of ZZ Top’s Tush. Not better but still....
Janis Joplin’s version of "Me & Bobby McGee" is imo better than Kristofferson’s (the song's writer. Never heard the Roger Miller version). But then Gordon Lightfoot flipped that again with his version. I witnessed Bill Johnson hearing Gordon’s recording for the first time (on LP in ’73), and being very impressed. Not just musically good, but sonically as well.
Starless, original by King Crimson, cover by The Unthanks on their album Last. OK maybe not better but nevertheless an interesting alternative take on a classic.
Nina Simone's 'Lilac Wine' is stunning. Jeff Buckley's cover brings me to my knees.
Aiofe O'Donavan singing "Nebraska"
Joan Baez The Night They Drove Old Dixie Down.....a tie with The Band’s Version with Levon
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.
Another toss up but...
by Percy Sledge or Lou Ann Barton