After reading here about Mariah Carey covering a Def Leopard song, I started thinking about what song(s) I would want to try to cover.
The first one I thought of was:
I will always love you By Whitney Houston. Done in punk style of screaming into the microphone.
What a great question! If only I could sing!
Jennifer Holliday’s And I Am Telling You (I think that’s what it’s called) from Dreamgirls!
I want to do a death metal cover of Hall & Oates “Maneater”. It’s such an awful song that there’s no way I could make it worse. I can do the vocals, but I don’t have a band, so no dice.
Hmm…I’ve often thought about this. I’ve been trying to get a band going for years with no luck yet. : ( Maybe while I’m here, everyone around here wants to be in a band it seems. But anyway, I’d want to cover:
“Girls Just Want To have Fun” - Cyndi Lauper (I was about 4 when that song came out and loved it back then)
“A Hard Day’s Night” - The Beatles
“Island in the Sun” - Weezer (because it’s a fun song)
Me and a friend did a sort of joke version of Hole’s “Garbageman” a few years ago that actually turned out ok. I didn’t know I could scream like that! I don’t have the tape anymore and I don’t think she does either, oh well.
I always wanted to cover the Eurythmic’s “Sweet Dreams” but Marilyn Manson beat me to it a bunch of years ago.
I’ve always fantasised about doing a techno house and hip-hop version of Jesus Christ Superstar.
Frank Sinatra singing the Nelson Riddle arrangement of I’ve Got You Under My Skin.
There is no other song!
Well, there’s also The Summer Wind, isn’t there?
I’d sing, perform, and arrange a male vocal version of Rufus And Shaka Khan’s Sweet Thing.
For some reason, I’ve always had a soft spot for Yazoo’s Only You.
“Sympathy for the Devil”, of course. 
No other option for an amoral young man like myself.
Oh, and maybe “Dirty Deeds”.
“Somewhere Over the Rainbow”, absolutely my hands down pick
Second pick is Peter Gabriel’s “In Your Eyes”
Me First and the Gimme Gimme’s just did a punk version a few years ago.
In college I played with a group of guys and we did nothing but acoustic covers of rock songs. By far our best translation was November Rain by Guns and Roses. It is the only one I prefer our sound to the original.
Playing the drums on “Wipeout”
“Up The Down Escalator” by the Chameleons. It’s just such a great song, and seems like it would be easy enough to do without making a mess of it.
“Rockin’ Girls” or “Eaten By the Monster of Love,” both by Sparks. And “Runaround Sue” by Dion.
This may have something to do with why I’m not in a band.
“All The Way”
“Stormy Weather”
If I had singing talent? Pretty much everything. I’ve discovered that I like singing a lot, I just can’t find a lot of songs I do well. That’s what you call a conflict of interest. 
Probably more Radiohead, since Thom Yorke is a tenor and sings so much higher than I can. There’s not a lot of rock stuff written for guys who sound like me, but most of it’s easier to do that THAT.
I can’t pick just one. But my current favorite singing in the car song is “Gloomy Sunday.” It’s just a tad high for me, though.
“I Can’t Stand the Rain” - heard an amazing version done by an a capella group in college, have wanted to sing it since.
“Once Upon a Dream (from Disney’s Sleeping Beauty)” - I know, sounds strange, but I’ve always thought of doing it with a punchier beat.
Susan
Black Hole Sun by Soundgarden. Chris Cornell’s vocals are unbelievable.
This is from a wannabe guitarist’s perspective, if I had the talent I’d play…
Hocus Pocus - Focus
Hey Lawdy Mama - by Steppenwolf has some amazing guitar work!
Star Spangled Banner - Jimi Hendrix
Eruption - Eddie VanHalen
Killing Floor - Jimi Hendrix 
Johnny B. Goode - Chuck Berry