If you just mean a song that was redone by a modern artist, then Lorde's cover of "Everybody Wants to Rule the World" is to die for! They just recently used it in the new Assassin's Creed: Unity trailer, but they also used it in the Hunger Games.
Her rendition of the song turns it from a silly and cheesy 80's song, to a powerful ballad and feels like stead fast rebellion turned into song. I'm really in love with this song.
If you mean a band who regularly redo's modern songs to sound like something else, then may I direct your attention to "Scott Bradlee and The Postmodern Jukebox"?
(Whoops, you get two songs because I love them both.) They redo modern songs (Like "Fancy" and "Talk Dirty" linked above) and turn them into completely new songs! They took "Fancy" and redid it in a "1920's Flapper" style, and it sounds amazing! With "Talk Dirty", they redid it in a sort of traditional Yiddish style, with the 2Chainz raps actually in Yiddish! Their version really brings a whole new life to that song!
They also did an amazing "1920's Gatsby" version of Lana Del Ray's " Young and Beautiful".
A couple of their covers sort of fall flat though. Their Bluegrass style cover of "Blurred Lines" is just hilarious, and their Gothic RB/James Bond cover of "Dark Horse" just sounds so meh to me.