Covers that are arguably better than the originals:

"I'd have to give a vote to Tori Amos' version of Losing My Religion (originally R.E.M - not that anyone in the developed world could fail to know that already) - quite possibly the most melancholy song ever (re)composed"

Wow -- I've never heard that one.

I think Tori's cover of "Famous Blue Raincoat" is better than the original. (ps--there are 2 you tube videos of tori's cover, one is lousy. The version I like is on a cover album of Leonard Cohen songs called "Tower of song - the songs of Leonard Cohen".)

JJ
 
Your Cheatin' Heart - Kitty Wells

Carolina in My Mind - Melanie Safka

All Along the Watchtower - Hendrix

Not that there's anything wrong with the original Hank Williams or James Taylor or Bob Dylan versions, just... most covers are nowhere, but not these.

w
 
I had mentioned this on other posts.
But I fell in love with "country delight" by Country Mike.
Mike D of the Beastie Boys pretending to be in a country band.
It's the polar opposite of the original, but still has a fun party vibe to it like the original "Rappers delight" by Sugar Hill!!!

Fischerspooners' version of the Buzzcocks "The 15th" is pretty awesome...

Visage's version of Zager and Evan's "In the year 2525" is mostly synthesizer and drum machine based (late 70's early 80's synth pop) so it actually feels more futuristic than the original and also maintains the melancholy feel with more dreary vocals 😀 Adding the fact that every remake of this song I have heard is frickin horrible, I'd say that this remake has stood the test of time rather well. The 60's folk social science fiction message just doesn't fit the sound of the original song.
 
I don't know that you'd call it better than the original, But Little Roger & the Goosebumps' "Stairway to Gilligan's Island" is probably the weirdest cover of the old chestnut, with words from the TV theme song set to "Stairway". Worth a listen (U-Toob does it), definitely good for a laugh or six.