Douglas Harper's Etymology Dictionary
fly-swatter
in reference to a bit of wire mesh on a handle, 1917, from fly (n.) + agent noun from swat (v.). Simple swatter was used in this sense by 1906. Other older names for similar implements were fly-duster (1860), fly-whisk (1836), fly-brush (1823), fly-fan (1821), fly-flap (mid-15c., glossing Latin muscarium).