I may come back and add more, but here's what leaps to mind:

Exit Music (For a Film) - Radiohead
Stockholm Syndrome - Muse
Five Years - David Bowie
Doghouse - No Doubt (I don't mean this to be about a pet kink, really! It's more an interesting relationship and perspective I think)
The Noose - A Perfect Circle
Pet - A Perfect Circle
Commissioning a Symphony in C - Cake
The Distance - Cake
Layla - Derek and the Dominoes
Layla (live acoustic) - Eric Clapton (these two are so different I'd consider nominating them separately, maybe, but one could nominate the first and consider the second, which has a very different take on the song :D)
The Dark of the Matinee - Franz Ferdinand (...this song is basically a hilariously hipster tale written long before hipsters were a solid cultural construct, but I like the story in it!)
Two Sisters - Andrew Bird (I'd love to see someone really dig into this mythos and creep everyone out)
Not a Robot, But a Ghost - Andrew Bird
Sleep - My Chemical Romance (This makes sense in the context of the album, generally, but I'd like to hear more about this guy's psychological state and suffering, because I'm awful like that)
You Know What They Do To Guys Like Us In Prison - My Chemical Romance (again: the narrator here is clearly in too deep and losing his shit, and I'd love to see someone tackle that)
Hang 'Em High - My Chemical Romance (WESTERNS)
Karma Police - Radiohead (could be awesome for a dystopia)_
No Surprises - Radiohead (another "what the hell is up with this cryptic person anyway" song)

And that's all I have for now. I don't recall if albums are an option, and this would be even weirder if it was, but I'd be interested to see what people could make of the pairing of Radiohead's albums OK Computer and In Rainbows, as explained here and listenable here.

But that's totally insane, and it's late here and I should probably post this comment before I come up with something even weirder and more convoluted. :P