I'm 75 and my grandmother sang this to me as a toddler on her lap. My family called it "Dear Mother". I want to thank our Library of Congress for pointing me to Mudcat, and all the folks who make Mudcat possible. I had been searching for the origins of this song for decades without success. I recorded our version and posted it on youtube (see link below). I apologize for my old thinning out voice.

Our family's version of "A Pretty Moon", we call it "Dear Mother"