Here are a few to get things started:
Don Henley’s The Boys of Summer as an 80s synth tune
Bossa nova version of Crowded House’s Don’t Dream It’s Over
Reggae-ish version of Aretha Franklin’s Respect
A capella men’s ensemble version of Lady Gaga’s Bad Romance
Yes – America (original by Simon and Garfunkel)
Anthony Vincent – Chop Suey in the style of Ghost (original by System of a Down)
Dr Pez & Marc Papeghin – The soundtrack of Ocarina of Time, but its a prog rock concept album (not sure it technically counts as a cover)
Brass Against – The Pot (Original by Tool)
The Jimi Hendrix Experience – All Along the Watchtower (Original by Bob Dylan)
Also, not sure how widely known this is, but Arethra Franklin’s Respect is itself a cover (and certainly one of the best covers of all time, up there with All Along the Watchtower and Johnny Cash’s rendition of Hurt)
Aretha’s version of Respect is so good that it’s essentially the de facto original, to the point that Otis commented “that a girl took away from me, a friend of mine, this girl she just took this song”.
To be fair, it makes so much more sense from a woman’s perspective.