The story
Possibly a variant of potato; compare uses like hot patootie (similar to hot potato) and sweet patootie (sweet potato). However, the adjectives hot and sweet are often collocated with words for the buttocks (regardless of which such noun), which limits the strength of that comparison; meanwhile, it is also possible that the similarity to the words toot and tooter may not be coincidental.
Etymology from Wiktionary, CC BY-SA 4.0.