"long, narrow strip of dried dough," 1779, from German Nudel, which is of unknown origin. West Flemish noedel and French nouille are German loan-words.
"improvise or play casually on a musical instrument," 1937 (implied in noodling), from noun meaning "improvised music," 1926, perhaps from noodle (n.), on analogy of the suppleness of the food and that of the trills and improvised phrases in jazz improvisations. Related: Noodled.
"head," early 15c., nodel, "head, back of the head," perhaps from Latin nodulus (see nodule). The meaning "simpleton, stupid person" (1753) probably is derived from it. Noodle-thatcher was old slang for "wig-maker" (1716).