late 14c., "flax prepared for spinning," also "refuse of flax used as kindling," somehow from the source of Old English lin "flax" (see linen). Perhaps from or by influence of French linette "grain of flax," diminutive of lin "flax," from Latin linum "flax, linen;" Klein suggests from Latin linteum "linen cloth," neuter of adjective linteus.
Later "flocculent flax refuse used as tinder or for dressing wounds" (c. 1400). Still used for "flax" in Scotland in Burns' time. Applied to stray cotton fluff from 1610s, though in later use this is said to be American English.