Kaineus

It might possibly be best to see this name as a masculinization of Kainis. After all: in Greek mythology, Kaineus and Kainis are one and the same person. Kainis was their original name, but it was changed to Kaineus after they were transformed into a man (at their request).

However, at the same time, we must consider the possibility that the etymology of their new name is different from Kainis. As in: it might refer to something more symbolic (rather than literal) about their transformation from woman to man. Such as a new beginning, or something along those lines. In that case, it would make sense for Kaineus to actually be derived from the Greek adjective καινός (kainos) meaning "new, fresh". Also compare καινέω (kaineo), which is thought to be a corruption or misspelling of the Greek verb καινίζω (kainizo) meaning "to make new or strange".