Related to Kinsey's Sexuality Scale. The Kinsey Report was written in the 1940s and 50s which found that there are more than the presumed 3 levels of sexuality (straight, bisexual, or gay), but that there are a multitude of sexual orientations. The report came up with a sliding 0-6 scale, defining "0" as completely heterosexual, all the way up to "6" as completely homosexual. A byke is a woman who would identify around a 4-5 level on the scale.
Derived from the two words "bisexual" and "dyke" = byke. Byke is sometimes used as a comical reference to deny the rigid societal regulations of sexuality. A related word that a woman might also use is "queer" in order to relate with the LGBT community.
Sara: well, your sexuality is up to you, ya know. wouldn't you agree that you are basically a lesbian, or dyke. we could call you a byke - basically you're gay but sometimes you like guys...not full on bisexual.
Anna: you're a dirty bi!
Callie: naw, im a Byke! i love the ladies...i connect with women almost always. but i reserve the right to occasionally date a man.
Anna: Ha...ok. i'll give ya that.