I make stuff up. I also alter the lyrics and use melodies that are familiar and easy (in my case, often Chrismas songs).
As a parent, I'm not wild about extra exposure to already dominant paradigms I don't agree with, but I very actively and seriously take intentions into consideration and will tolerate quite a bit from people I love who have good intentions. :)
You, being actually a good singer, could probably make up WAAAAY better stuff than I do. (Although sometimes just "go to sleep, go to sleep, go to sleep little one" is all one can manage!)