Was (Not Was) had a song called “Zaz Turned Blue,” which they arranged for Mel Torme to sing.
It was about a guy named Zaz who got into a mock fight in the park with his friend Steve Brown. Steve pretended to strangle Zaz, who turned blue and nearly died of asphyxiation. Zaz survived, but seems to have come out of the experience mentally handicapped (he goes around constantly with “a silly grin on his chin”).
“Birdhouse in your Soul” by They Might Be Giants. It’s about a nightlight.
TMBG should probably fall under the “Weird Al Rule” for purposes of this thread. 
*Ms. Otis Regrets *is a 1934 Cole Portersong, in which Ms. Otis comes to a bad end after a love affair, kills her lover and is killed in turn by a mob. The refrain is her sending her regrets about missing her lunch date, as she’s a civil high-society type lady.
Absolutely tons of these from musical theatre:
I could go on, but we’d be here all night.
Beat me to it. I would also add “Play It All Night Long.” Dysfunctional country family, incest, PTSD, etc. I’m pretty sure Warren Zevon is the only person to use the word “Brucellosis” in a song.
Also, “Exciteable Boy.”
After ten long years they let him out of the home
Exciteable boy they all said
Then he dug up her grave and built a cage with her bones
Exciteable boy they all said
Raspberry Swirl by Tori Amos is about women forming emotional bonds with each other, but the title references oral sex
If you want inside the well, boy
You’d better make her raspberry swirl
Many Rush songs probably fit the bill but my favorite one is Manhattan Project which is about… the Manhattan Project.
Let me just say, anything by Ray Stevens. “Ahab the Arab,” “The Streak,” etc. If it’s by Ray Stevens, it’s an unusual topic (if you think about it, “All the World Is Beautiful” is even unusual [in its own way, of course])
“Does Your Bubblegum Lose Its Flavor On the Bedpost Over Night?” about bubblegum losing it flavor on the bedpost.
“Mares Eat Oats” about quadripeds eating oats and ivy.
“Please Mr. Custer,” about a soldier going out with General Custer on his last foray in the field and the soldier not wanting to be there.
If you think about it, the Beach Boys’ “In My Room” is an ode to being alone in a room…Kind of weird.
OMG, I don’t even know what to say. Wow
You kinda have to take it in the context of the rest of the musical, if you’re not familiar with it. It’s not quite as bad as it seems (though I read in a book about the musical that the actors who had to sing it had a lot of issues at first, but once they got into the show and saw the audience reaction, they got more comfortable with it.)
After reading the other entries in this thread, this one seems kind of tame, but I’m still surprised no one has mentioned Michael Jackson’s “Ben,” which is a love song sung by a young man to his favorite killer rat.
Also, there’s “The Vatican Rag,” from the movie, “The Ruling Class.” I’m afraid I can’t remember the details. I’ve got to see that movie again someday.
“The Crablouse” by Lords of Acid is about…crab louse and specifically how the singer enjoys having them.
Many of mine were already taken about, but how about Being Boiledby (early) The Human League? Any song that starts:
Listen to the voice of Buddha
Saying stop your sericulture
is not your average everyday pop song. Others from that album include “A Crow and a Baby” (had an affair …), and “The Black Hit of Space” (it went to number one, then into minus figures…).
Presumably this is the Tom Lehrer song by the same name?
“Stay on the Ride” by Patty Griffin is about Little Old Man, who rides the bus every day, but without going to any destination or even knowing where the bus goes. This puzzles the bus driver, who asks him about his aimless riding. Little Old Man’s advice is: “Stay on the ride, it’s gonna take you somewhere.”
“Spider Web” by Joan Osborne is my favorite song ever. She sings how she has a dream that Ray Charles got his eyesight back, but Ray don’t sing no more. Then he takes his glasses off and she can look inside his head, and she sees a shining spider web. He takes her flying in the air and shows her own spider web, and tells her the world is made of spider webs: “the threads are stuck to me and you.”
NaCl (The Salt Song) by the McGarrigle Sisters. About a “little atom of chlorine … swimming through the sea just having fun”. “But somewhere in that sea lurks handsome sodium…”
Loudon Wainwright III (ex-hudband of Kate McGarrigle) had a chart success with Dead Skunk in the Middle of the Road.
Tom Waits might need a similar category… Step Right Up is an extended sales pitch; The Ocean Doesn’t Want Me Today is about someone NOT committing suicide… today; Pasties and a G-String is about strippers; Frank’s Wild Years is about a little Chihuahua named Carlos with some kind of skin disease… and arson (never could stand that dog).
Yeah, I’m thinking separate category.
(YouTube link–song lyrics are very NSFW)