I don't know if that would count as a cover. But French singer Johnny Halliday did an French adaptation of Los Bravos' hit "Black is Black" as "Noir c'est Noir".
Sam Smith covers Whitney Houston, I like how melancholy the cover is from an upbeat pop song. Really changes the meaning, or shows how dark the lyrics really are.
Kings of Leon cover Dancing on my own.
A great cover of Africa, honorable mention. That high part is really hard! I've eaten at this particular pizza joint in SLC (The Pie) wish I could have heard this live, stop by if you're ever in east Salt Lake City.
Nouvelle Vague is a French group that made their name by bossa-nova or downtempo cover of 80s songs, especially synth pop and goth stuff. Of late they have created original songs.
This one reminds me of the Australian band Frente's cover, who had its heyday:
A band from my metalcore days covered
Smashing pumpkin's today. Ngl I still like the sound and Poison the Well is the best band from my angsty teen years
I mentionned on the Oldies thread but Love Hurts first performed by The Everly Brothers and then Roy Orbison as a soft romantic song, got eclipsed by the cover done by Nazareth who did it as a power ballad. It's hard now to think of the original version done by the Everly Brothers because it's the cover done by Nazareth who come first in mind.
This is from an album produced waaaaay back in 1987: "Stay Awake: various INTERPRETATIONS of Disney songs" They are not covers but unique personal intepretations of songs by various artists such as Tom Waits and his interpretation of the 7 dwarves Song "Hi Ho"