"handle of a mop," 1710, from mop (n.) + stick (n.).
Entries linking to mopstick
mop n.
late 15c., mappe "bundle of coarse yarn, cloth, etc., fastened to the end of a stick for cleaning or spreading pitch on a ship's decks," perhaps from Walloon (French) mappe "napkin," from Latin mappa "napkin" (see map (n.)). Modern spelling by 1660s. General sense, of such an implement for cleaning floors, windows, etc., is from 1610s. Of smaller utensils of the same sort used for cleaning dishes, etc., by 1869. Of anything having the shape or appearance of a mop (especially hair), by 1847. Grose ["Dictionary of the Vulgar Tongue," 1788] has mopsqueezer "A maid servant, particularly a housemaid."
stick n.
Old English sticca "rod, twig, peg; spoon," from Proto-Germanic *stikkon- "pierce, prick" (source also of Old Norse stik, Middle Dutch stecke, stec, Old High German stehho, German Stecken "stick, staff"), from PIE root *steig- "to stick; pointed" (see stick (v.)).
Meaning "staff used in a game" is from 1670s (originally billiards); meaning "manual gearshift lever" is attested by 1914. Alliterative connection of sticks and stones is recorded from mid-15c.; originally "every part of a building." Stick-bug is from 1870, American English; stick-figure is from 1949.