How about "Rockabye Sweet Baby James" by James Taylor?
(I used to wonder why JT was writing and singing songs about himself being rocked to sleep. Then I learned the song was written for his baby nephew or someone like that. LOL )

There's also a cute Yiddish lullabye called "Yankele," which I have on a Theodore Bikel album.
The mother is trying to get her baby boy to go to sleep, singing to him about how he's going to grow up to be a Talmudic scholar or a famous lawyer, but why must he keep wetting his bed?