I don't know a kid from the 1980s who can't hum this song, and those of us who took piano lessons most likely played it at a recital as well. But I learned more about the song from the Ask Metafilter question.
Someone pointed out that it is often played by ice-cream trucks. (I think the one in my neighborhood plays "Reuben, Reuben" but it may have a variety of tunes.) Another person pointed out that there are at least two versions -- the Frank Mills hit (video below) and one by Richard Clayderman. (Frank Mills, by the way, has nothing to do with the awesome "Hair" hit of the same name. I like the Lemonheads cover.)
Also, I had no idea there were lyrics. But I had the music box -- didn't every girl? And for just $10 at Amazon, I may have to buy one for Kelly sooner rather than later.