I'll lay out some of my favorites:
Dead Kennedys' I Fought The Law (originally by Bobby Fuller). They reinvented the song to call out the murderer of Harvey Milk for getting away with...Yanno, murder. And it just sounds good! Jello Biafra probably has one of my favorite voices in all of music ngl and he does it well here. I could also put in there cover of the theme to Rawhide, though that'd mostly be because Jello sounds so excited to be a cowboy!
DEVO's cover of Satisfaction (originally by the Rolling Stones). They straight up made it their own and it feels so distinctive and...Different. It's awesome, and there was even a story told by one of the lead singers of DEVO, Jerry Casale, who claimed that when Mick Jagger heard their cover, he straight up danced it out! Of course, the account changes each time and the validity of the statement could be argued but I believe it because I mean...Listen to it. It gud.
Oingo Boingo's Violent Love (originally by Willie Dixon I think?). I actually didn't know it was a cover of theirs, like the lyrics and way they did it seemed so fitting for them. Good song, yo!
Lard's cover of They're Coming To Take Me Away (originally by Jerry Samuels). Jello is back and is going insane, as he should be. Not much more to say, other than this song fits Jello like a glove.
Jack Off Jill's cover of Lovesong (originally by The Cure). I LOVED the original song so when I first heard JOJ's cover, I was blown away! It's got the same heart, yet it feels so different. I dunno, it just screams "bad girl falls in love with you" vibes~ Then again, I was in high school so it's also nostalgic to me. Just like how The Cure's original is nostalgic to me, albeit for different reasons.
I probably have more but my brain don't work good at the moment so that's all I came up with. Also if parodies of songs count then Weird Al is a good contender, but then it'd be like half his discography and it'd never end lol