“Hole at the bottom of the sea.”
(“There’s a flea on the fly on the wart on the frog on the bump on the log in the hole at the bottom of the sea…”)
If you have time (because they’ll need lyric sheets), kids LOVE “The Cat Came Back”
My boys like “Big Rock Candy Mountain”, the old folk song. There is an original version talking about cigarettes and whisky, and a newer version replacing the adult things with lemonade and soda pop.
99 Bottles of beer on the walk.
Take one down, pass it around
98 bottles of beer on the walk
kids gotta learn it sometime. 
That starts with “Never laugh when a hearse goes by / For you may be the next to die”. That might make it a little easier to find.
Along a similar theme, there’s “Gory Corey” (to the tune of “Battle Hymn of the Republic”):
…plus about ten more verses
Or Tom the Toad (to the tune of O Taunnenbaum)
What I find hilarious is that the girl scouts have books chock-full of gross songs about animals getting hit by cars–and the boy scouts had something like six songs total.
My pick:
There’s a hole in my bucket
Dear Liza, dear Liza
There’s a hole in my bucket
Dear Liza, a hole
Then fix it dear Henry
Dear Henry, dear Henry
Then fix it dear Henry
Dear Henry, fix it
The rest of the lyrics can be found at its Wikipedia page.
There’s a skeeter on my… Uh. maybe not.
Thanks for the (serious and not so serious) replies. I went with “There Was an Old Lady Who Swallowed a Fly”. Went over well.