1702, of unknown origin, perhaps a metathesis of film (n.) "gauzy covering" + -y (2). Figuratively (of arguments, etc.) from 1750s. Related: Flimsily; flimsiness.
flimsy construction
a flimsy table
a flimsy argument
flier
flight
flightless
flighty
flim-flam
flimsy
flinch
flinders
fling
flint
flint-lock