A Greek person is called an Ἕλλην. Did the Greeks have a word for “Greek language” in Classical times? With my Middle Liddell, it looks like Ἑλληνικός is the word for Hellenistic Greek, but I am uncertain if Classical Greek had a word for it. Maybe they used the verb **Ἑλληνίζω?**Perhaps they thought of themselves as speaking Attic, Doric, Ionian, Aeolic, or some other dialect?
Was that your question, or were you after a substantive? (If you’re after this, see e.g. Plato Cratylus 410a: τοῦτο γὰρ οὔτε ῥᾴδιον προσάψαι ἐστὶν Ἑλληνικῇ φωνῇ.)