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" (1883).
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 1630s.
The Dictionary of American Slang, Fourth Edition by Barbara Ann Kipfer, PhD. and Robert L. Chapman, Ph.D. Copyright (C) 2007 by HarperCollins Publishers. Cite This Source
Idioms and Phrases with by the bye
by the by. Incidentally, in passing, as in By the bye, my wife is coming too, or Exactly where do you live, by the by? The bye or second by in this term originally meant “a side path,” whence the current sense of “off the track” or “of secondary importance.”
[ Early 1500s