Английский Этимологический словарь - bye

- variant of by (prep). In cricket, "a run scored on a ball that is missed by the wicket-keeper" (1746), hence, in other sports, "position of one who is left without a competitor when the rest have drawn pairs" (1883). Also, a shortened form of good-bye. Reduplication bye-bye is recorded from 1709, though as a sound used to lull a child to sleep it is attested from 1636.