Phineus was the name of a brother of Cepheus, the uncle of Andromeda, and of the blind soothsayer of uncertain parentage who became the king of Salmydessus, in Thrace. Although the name is quite well known, and occurs both in the Bible and in literature, as well as being a historical name and a rarish modern first name, its origin is unclear.
There are no words starting phinin Greek, so all we can do is to consider the same sea-bird as for Phineïs (which see) or some such origin as phthino, 'to decline', 'waste away'. But there is no evidence that either of the two characters mentioned here did that.
Other derivations for the name in general that have been proposed are an Egyptian word meaning 'dark-complexioned' and a Hebrew word meaning 'mouth of brass' or 'oracle'. Harry Long derives the name from Hebrew pi, 'mouth' and nahash, 'hisser', so that Phineus is a 'snake-mouth' or tempter.