"kick," 1845; see punt (v.).
"flat-bottomed river boat," late Old English punt, perhaps an ancient survival of British Latin ponto "flat-bottomed boat" (see OED), a kind of Gallic transport (Caesar), also "floating bridge" (Gellius), from Latin pontem (nominative pons) "bridge" (see pontoon). Or from or influenced by Old French cognate pont "large, flat boat."
"to kick a ball dropped from the hands before it hits the ground," 1845, first in a Rugby list of football rules, perhaps from dialectal punt "to push, strike," alteration of Midlands dialect bunt "to push, butt with the head," of unknown origin, perhaps echoic. Student slang meaning "give up, drop a course so as not to fail," 1970s, is because a U.S. football team punts when it cannot advance the ball. Related: Punted; punting.