It derives from the ancient greek word Ἑλλην-ιστής or "Hellénistés", meaning anyone who speaks the Greek language.

A German historian named Johann Gustav Droysen started using the word to refer the period when Greek culture spread in the non-Greek world after Alexander’s conquest.

Well, Thucydides states in Book 1: Before the Trojan war there is no indication of any common action in Hellas, nor indeed of the universal prevalence of the name; on the contrary, before the time of Hellen, son of Deucalion, no such appellation existed, but the country went by the names of the different tribes, in particular of the Pelasgian. It was not till Hellen and his sons grew strong in Phthiotis, and were invited as allies into the other cities, that one by one they gradually acquired from the connection the name of Hellenes; though a long time elapsed before that name could fasten itself upon all. The best proof of this is furnished by Homer. Born long after the Trojan War, he nowhere calls all of them by that name, nor indeed any of them except the followers of Achilles from Phthiotis, who were the original Hellenes: in his poems they are called Danaans, Argives, and Achaeans.