I’m guessing most people don’t even realize (There’s) Always Something There to Remind Me was a cover when Naked Eyes released their version.
Slade was never a big hit in the US but some of their songs were. I always preferred Noddy. They did come back and have a couple of minor US hits in the 80s.
1984 brought us this…
How many have recorded it since then? My favorite is John Cale’s version on Shrek. You can buy the Shrek soundtrack but it somehow gives a Rufus Wainright version. You can buy a Scrubs soundtrack that has John Cale singing it, but it’s not the version from Shrek. I gather you have to watch Shrek to hear my favorite version.
Was Tiffany’s version a bigger hit than Tommy James’?
I don’t know why Columbia Records couldn’t produce a hit single for Laura Nyro, when she was putting out studio work like (just to name three) Eli’s Coming
Wedding Bell Blues
Stoned Soul Picnic
Tommy James and the Shondells made it to #4 on the Billboard Hot 100 chart with it; Tiffany did actually outperform that, reaching #1. But, it was certainly a major hit for both of them.
My fave remains:
And Save the Country, another big hit for the Fifth Dimension.
you could mention always on my mind originally written by willie nelson it was something of a hit for Elvis Presley but has had numerous covers by the pet shop (in fact they seem to make a remix duet or live version of it at least once a year and he has thanked "those lads in England"for their support on more than one occasion
i hope Neil and willie sing it together at least once before willie passes
Carole King wrote and recorded “You’ve Got a Friend” for her album Tapestry, and was inspired to write it by a line from her friend James Taylor’s song “Fire and Rain.”
King was recording her album at the same time that Taylor was recording his album Mud Slide Slim and the Blue Horizon, and the two of them were working with the same backup musicians. Taylor asked King if he could cover the song on his own album, to which she agreed.
King didn’t release the song as a single in the U.S., but Taylor’s version became a #1 hit. I like Carole’s version better, but I admit that I’m a Carole King fanboy. 
A 5th of Beethoven
Joan Baez, 1of the finest of singer/songwriters, wrote & recorded “Diamonds And Rust” for her 1975 album of the same name. Musically and lyrically, a very intriguing song about her relationship with a guy by the name of Bob Dylan.
So one would think that a lot of other acoustically styled musicians would cover this song & I’m sure they did. But who had perhaps the biggest hit with it? None other than the heavy metal gods Judas Priest, a staple of pretty much every concert they played. Who’duh thunk it?!? 
Steve Goodman’s City of New Orleans:
Arlo Guthrie wisely decided to slow it down.
The Pat Boone version is just awful isn’t it?
Along similar lines - Joni Mitchell’s 'This Flight tonight
You’d not really imagine this could become a bluesy rocker - but you’d be wrong, and it was a huge hit by Nazareth
And Nick Rivers performed it for yet another audience…
Let’s talk to the Boss. Mannfred Mann did ok with these.
When I wrote the OP I was thinking about artists listening to the albums others put out and thinking, ‘We should do that.’ Then it would be fun to see what treatment they gave it, imagine how they got from A to B. Speed it up? Add horns? Go minimalist, with just a piano or guitar maybe? Fuse it with another genre, maybe taking a rock song and adding country elements, etc. But some don’t stray far from the source material.
I gather that there are songs Springsteen wrote (Fire/Pointer Sisters) or co-wrote (Because the Night/Patti Smith) and he maybe recorded them, but they didn’t make the cut. They didn’t show up in concerts or records until others had made them hits.
More about his songs…he wrote “Fire” thinking it would be a good one for Elvis??
The mention of Bruce’s “Fire” reminded me that, after June Carter and Merle Kilgore wrote “Ring of Fire,” it was initially recorded by June’s sister, Anita. Johnny Cash loved the song, and after it wasn’t a hit for Anita, he recorded his own version of it, which, of course, became one of his signature hits.
Has anyone mentioned Whitney Houston’s cover of Dolly Parton’s “I Will Always Love You”? Thats a major one for both.
Jazz is literally chock-a-block with that sort of music, often ruining it in the process. That’s especially true when a jazz vocalist turns a really good song into a lounge tune (remember Bill Murray’s bits?), thus removing whatever soul the song had.
I don’t know if that is universally true. Holly Cole did an excellent cover of “I Can See Clearly Now” that was well received.
