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?

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

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.

Bizarre Love Triangle. The original is some very bad pop music. New Years Day brought it not only to rock, but into being really good too.

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

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.

“House of the Rising Sun” has been covered numerous times. The most known version is from The Animals. I was today years old when I learned that this is the original version of the song. However, my personal favourite cover version is from BTO.

Another well-executed cover version is of the song “Johnny B. Goode”, covered by Peter Tosh, beause it is tranferred into a different music genre.

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

Stille Volk - Adoumestica Una Terro

“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.

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

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

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

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 Sound of Silence by Disturbed.

Original. - had no much success until the cover.