First note that the Greeks do not call themselves Greeks. They are Hellenes. “Greek” originally designates the Hellenes of Southern Italy, and our use of this word comes from Latin “Graecus”.

During the whole Byzantine period, they called themselves Rhomaioi = “Romans”. The word Hellenes = “Greeks” specifically meant the pagans (some parts of Peloponnese, such as Mani peninsula I believe, had a significant Pagan population up until 10th century). “Greek” was used by Latins as an insult (you find it, for example, in Villehardouin's account of the Fourth Crusade), to diminish the Roman empire to a Greek kingdom.

During the late Byzantine period, once there were no Pagans to call Hellenes, it became a synonym for “Greek-speaking” - I think it was Anna Komnena who wrote something like "by language we are Hellenes, by religion we are Christian, and our Empire is Roman", but I cannot find my source right now).

During the Ottoman period, peoples under Ottoman rule were grouped by religion and not by language, under the millet system. Thus the Greeks were simply “Christians”.

At the beginning of Greek nationalism (late 18th century to early 19th century), there was a fierce competition between the names "Hellenes" and "Romans" as the new name of the Greeks. The name “Hellenes” won, in part because it was useful to evoke the memories of Classical Greece to foreign backers (Britain and France in particular), and in part because “Roman” was then seen as a symbol of the failed, “decadent” Byzantine empire.