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)