Asklemmy

Cyndi Lauper's Girls Just Want to Have Fun was originally Robert Hazard's Girls Just Want to Have Fun

Gloria Jones originally performed Tainted Love in the ‘60s. It didn’t hit until Soft Cell covered it in ‘81.

“Torn,” by Natalie Imbruglia (1997) was originally written by Ednaswap and recorded in 1995—which, imo, is better. However, even though they wrote it, it was first recorded in 1993 by danish band, Lis Sørensen, who titled it “Brændt,” which means “Burned.”

Bonus! "The State I'm In" a song from Ednaswap’s first album, was covered by Sinéad O'Connor for her 2000 album Faith and Courage.

Harry Belafonte's exuberent cover of "Day O (Banana Boat Song)" which was a hit in its time and re-popularized by the dinner scene in Beetlejuice.

The original is Jamaican folk/traditional and the first recording of it Eric Connor - Day O has a totally different tone to it. In the original, labouring on a banana plantation is as tiring and depressing as you'd think.

Whiskey In The Jar Song by Metallica, original is from ~~the Dubliners in 1969~~ a traditional Irish song in the 1950s

You want obscure? Here's a cover of a song by Joe Glazer, written in the 1940s by a union coal miner that immigrated to the US

The Amen Break is a few seconds sample that became drum and bass. All of drum and bass has roots from that sample.

Rage Against the Machine's Renegades of Funk was originally written and performed by a band called Afrika Bambaataa & Soulsonic Force in 1983.

RAtM

Afrika Bambaataa

Not sure if this applies, but people still seem to think that Nine Inch Nails covered Hurt by Johnny Cash and not the other way round. That or they haven't even heard the original.

probably The Association's cover of "Windy" by Ruthann Friedman (which she wrote about a guy, btw)

This cover by the Afghan Whigs of the song "Lost in the Supermarket" by The Clash is really good:

https://www.youtube.com/watch?v=AzDJhOMjq1E

The song 1985 was not originally recorded by Bowling for Soup, but SR-71

The Louis Prima Orchestra's cover of "Whatta Ya Gonna Do?", by Sunny Skylar. One of my favorite songs, a perfect blend of New Orleans and New York in the 40s. The link above is the only version I can find on the internet, which is instrumental. The version on the record I have has vocals and is cleaner. I hope somebody works to preserve Louis Prima's catalog from the 40s digitally, there's so much old music at risk of being lost!

Do obscure rap samples count? Because I love these two:

https://music.youtube.com/watch?v=Sy9nSEYZQzE&si=6x72ueIzb4LvqImv

https://music.youtube.com/watch?v=4TqyGMjeIpA&si=RkaU_sQJK24G9M9T

To the original question, I would pick RHCP cover of Sly and the Family Stone's song If You Want Me To Stay.

https://music.youtube.com/watch?v=jkzWSL-gEO4&si=l-LP8ose9welAblw

https://music.youtube.com/watch?v=jsXkP0HTf3w&si=OrswIwjoCLw1jHO

Also Type O's cover of Seals and Croft's song Summer Breeze

https://music.youtube.com/watch?v=ywL6tMQdG4c&si=9uM1HqkGh1rSwv7V

https://music.youtube.com/watch?v=uiKzKdLk-iQ&si=pQRsVqWN8iAUzSDn

Well apparently people think Darius Rucker wrote the Old Crow Medicine Show song Rock Me Mama, and what's more, OCMS in their turn credited it (in jest) to Bob Dylan.

Drives me crazy that people don't know Stomp and Holler is a Hayes Carll song, and that Hallelujah is a Leonard Cohen song, and Downtown Train a Tom Waits song, too.

But I don't know if any of these are as obscure as you are looking for. You are looking for songs even music people don't think of as covers?

once upon a band we played "Sister Andrea" by Mahavishnu Orchestra with many precisions and very many clarities ♫♪!! (it sounded pretty close)

John Coltrane's versions of Afro Blue. Very different from Mongo Santamaría's original. I like both approaches, but Coltrane's recordings of it were all just so powerful.

God Gave Rock and Roll To You was a hit for Kiss.

Original by Argent: https://m.youtube.com/watch?v=QsG5V-o6uxY&pp=ygUdZ29kIGdhdmUgcm9jayBhbmQgcm9sbCB0byB5b3U%3D

Type O's cover is so weird: really turns out on its ear

Johnny Cash's cover of Hurt. Trent came off a bit whiny. Cash... it's his song now.

Stille Volk’s cover of Iron Maiden’s To Tame A Land:

Stille Volk - Adoumestica Una Terro

Dax Riggs covering Bonnie Prince Billy's Grand Dark Feeling of Emptiness

https://youtu.be/obnh0-NOjZg?feature=shared

Two that are not at all obscure to but often overshadowed by their covers... i love the Four Seasons version of Beggin - https://youtu.be/d5AfvOk57bE?feature=shared

Same for the Everly Brothers doing Love Hurts - https://youtu.be/hFE2SnliiV0?feature=shared

Edit to add - i didn't know Peace, Love, and Understanding was Nick Lowe's.