bye (n.)

in sporting use, a variant of by (prep). Originally in cricket, "a run scored on a ball that is missed by the wicket-keeper" (1746); later, in other sports, "position of one who is left without a competitor when the rest have drawn pairs" (1868).

1709, shortened form of good-bye. Also compare lullaby. Reduplication bye-bye is recorded from 1736; as a sound used to lull a child to sleep it is attested from c. 1500.