early 14c., "little headband," from Old French filet "thread, filament; strip, ligament" (12c.), diminutive of fil "thread" (see file (v.1)). Sense of "cut of meat or fish" is from late 14c., apparently so called because it was prepared by being tied up with a string.
c. 1600, "to bind with a narrow band," from fillet (n.). Meaning "to cut in fillets" is from 1846. Related: Filleted; filleting.