what is the source for the word (bye)?

Bye may refer to:

Bye (cricket), a special type of run scored in the game of cricket

Bye (sports), when a player or team is allowed to advance to the next round of a playoff tournament without playing

Goodbye, a traditional farewell phrase

A surname, originally meaning to live on a bend of a road or river

“Bye,” a song by Elliott Smith from his 2000 album

Now here is what you probably really want:

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.

good-bye

1591, from godbwye (1573), itself a contraction of God be with ye, infl. by good day, good evening, etc.

Goodbye (or the shortened form, goodby, sometimes used in American English, or simply bye) is a traditional farewell phrase used in the English language. Most scholars agree that the phrase originally derives from “God be with you.” . It has also been suggested by some that it originated from “God buy [redeem] you”.

Goodbye is from old English for God be with you.