In 1578, a high lama of the gelug school was given the name dalai lama.” what does the sino-mongolian translation of dalai mean in english?

There are two meanings of the Sino-Mongolian translation of the word dalai in English - this word can mean both "ocean" and "big." In the context of Dalai Lama's title, it would be better to chose the latter meaning, given that the word Lama means "master, guru."
So the literal translation of the title would be - big master.