also quarter-staff, 1540s (quarter-stroke "stroke with a quarterstaff" is attested from early 15c.), an old weapon formed from a stout pole, six to eight feet long (six-and-a-half sometimes is given as the standard length), tipped with iron, formerly a weapon characteristic of the English peasantry. From staff (n.); the quarter in it is of uncertain signification. According to one theory, favored by fencing manuals, etc., it likely is in reference to operation of the weapon:
It was grasped by one hand in the middle, and by the other between the middle and the end. In the attack the latter hand shifted from one quarter of the staff to the other, giving the weapon a rapid circular motion, which brought the ends on the adversary at unexpected points. [Century Dictionary]
Linguists tend to prefer an explanation from woodcutting, perhaps a reference to a cut of lumber known as a quarter, but contemporary evidence is wanting for either conjecture.