In the hypostatic union, is Jesus just the name of the man or of the God or of both?

First things first: I am not a Trinitarian. However, I know many good people that are Trinitarians and some of them are very close to me so we'll say that I'm sympathetic to the cause. I'm sincerely interested in knowing the answer to this (and others) query to better my understanding. What led me to the question is the belief that many Trinitarians hold regarding the name Jesus: it only belongs to the Son. I understand this to be a way of avoiding modalism, i.e. Jesus is not the Father. However, the name Yahweh can apply to the Father, Son, or Spirit (correct me if I'm wrong). So is Jesus the name of the person of God in Him, or only the man, or of both God and man? In my mind it makes sense if it's only the man but I could be way off!