Hurray! another Kentuckian! I'd like to extend greetings to you from Lexington.

Anyway, "earworm" is what came to my mind too. The only French term even close that I know of doesn't really pertain to music: idee fixe (I probably spelled it wrong but there it is ), which I am pretty sure means an idea on which one becomes fixated. I suppose a tune is an idea in the most abstract sense but I think "earworm" is better.