From the column "Music in the News" by Margaret S. Wells, in Billboard Feb 12, 1944, page 65:

EIGHTH-NOTES.—A simple explanation of the meaning of the current bit of nonsense, Mairzy Doats, is given by "Kup" in his column in The Chicago Times. When Milt Drake, who wrote the song with Al Hoffman and Jerry Livingston, was asked how he happened to hit on the idea, he answered: "Kideech ants ong—far the rite sit—milyink oppees old." Translated from "mairzyology" into English, that means, he explained, "Kiddy chants song—father writes it—million copies sold." . . . Idea for the song first struck Drake more than two years ago when he heard his four-year-old daughter chanting something that sounded like "cowzy tweet and sowzy tweet and liddle sharsky [sic] doisters." That brought to mind a nursery rhyme he had learned as a child. Drake then contacted his two song-writing partners and Mairzy Doats (mares eat oats) was born.


[This anecdote has been repeated, more or less, in many publications. This is the oldest version I could find. I suspect the idea that the girl was repeating an existing "nursery rhyme" was an embellishment on the original story. More likely she made it up herself, or Drake made it up because he couldn't remember exactly what she said. Note the story says "something that sounded like...".]