"narrow strip of dried dough," 1779, from German Nudel, which is of unknown origin. West Flemish noedel and French nouille are German loan-words. The older noun meaning "simpleton, stupid person" (1753) probably is an unrelated word, as is the slang word for "head" (attested from 1914).
1937 (implied in noodling), from noun meaning "improvised music," 1926, probably from noodle (n.), on analogy of the suppleness of the food and that of the trills and improvised phrases in jazz improvisations. Related: Noodled.