Why'd they add the 'e'? Just one of a million quirks of the English language, I guess.
From
http://www.word-detective.com/101404.html :
Live and learn. I know I'm supposed to affect a pose of omniscience when I write this column, but every so often a reader writes in with a query about a word or phrase of which I have never heard, but which seems to be rather well-known in some precincts. In this case, it turns out that "bye" in the sense you use it is well-established in a number of sports, and has been in use since the mid-18th century.
The earliest uses of "bye" (also spelled "by") were apparently in cricket, where it means, according to the Oxford English Dictionary, "A run scored for a ball which passes the batsman, and which the wicket-keeper and long-stop fail to stop." I haven't the faintest idea of what that means, but the sense seems to be that a run is scored for a play which did not actually take place and the player advances without actually playing. This leads us to the more general sense of "bye" used in other sports, "The position of a player in a tournament who advances to the next round without playing, usually because there is an odd number of players." Skipping a game because of a lack of dart boards would, as in your case, also qualify as a "bye."
"Bye" in this sense is a specialized use of the common English preposition "by" as a noun meaning "something secondary" or "an incidental matter" as compared to the "main" thing. It's the same sense of "by" found in "byway" or "bypass" meaning a secondary road. So in a sense your "bye" simply means that you have "bypassed" playing a game this week.