While most Homerists (I think) accept the argument advanced by L. R. Palmer and Gregory Nagy that Achilles’ name (Akhilleus) derives from akhos (“woe”, “grief”) and laos (“host, people, army”), some ancient authors had different ideas.
Apollodorus. 3.172.4-5
“After Thetis gave birth to a child with Peleus, she wanted to make him immortal—without Peleus knowing, she used to cover him in fire at night to destroy his mortal inheritance from his father, and at day she rubbed him down with ambrosia. After Peleus discovered this and saw his son struggling in the fire, he cried out. And Thetis, prevented from completing her plan, abandoned her child and went to the daughters of Nereus. But Peleus took his child to Kheiron. Kheiron accepted him and fed him the innards of lions wild boars and with the marrow of bears. And he named him Achilles, because his lips (kheile) never touched [her] breasts. Previously, his name was Ligurôn.”
So the explanation here, is that his name is ‘alpha-privative’, meaning something like “Lipless”
ὡς δὲ ἐγέννησε Θέτις ἐκ Πηλέως βρέφος, ἀθάνατον θέλουσα ποιῆσαι τοῦτο, κρύφα Πηλέως εἰς τὸ πῦρ ἐγκρύβουσα τῆς νυκτὸς ἔφθειρεν ὃ ἦν αὐτῷ θνητὸν πατρῷον, μεθ’ ἡμέραν δὲ ἔχριεν ἀμβροσίᾳ. Πηλεὺς δὲ ἐπιτηρήσας καὶ σπαίροντα τὸν παῖδα ἰδὼν ἐπὶ τοῦ πυρὸς ἐβόησε· καὶ Θέτις κωλυθεῖσα τὴν προαίρεσιν τελειῶσαι, νήπιον τὸν παῖδα ἀπολιποῦσα πρὸς Νηρηίδας ᾤχετο. κομίζει δὲ τὸν παῖδα πρὸς Χείρωνα Πηλεύς. ὁ δὲ λαβὼν αὐτὸν ἔτρεφε σπλάγχνοις λεόντων καὶ συῶν ἀγρίων καὶ ἄρκτων μυελοῖς, καὶ ὠνόμασεν ᾿Αχιλλέα (πρότερον δὲ ἦν ὄνομα ὠνόμασεν ᾿Αχιλλέα (πρότερον δὲ ἦν ὄνομα αὐτῷ Λιγύρων) ὅτι τὰ χείλη μαστοῖς οὐ προσήνεγκε.
Kallierges, Etymologicum Magnum 182
“Akhilleus: [this name comes from] lessening grief, for Achilles was a doctor. Or it is because of the woe, which is pain, he brought to his mother and the Trojans. Or it is from not touching his lips to food [khilê]. For he had no serving of milk at all, but was fed with stag-marrow by Kheiron. This is why he was hailed by the Myrmidons in the following way, according to Euphoriôn:
He came to Phthia without ever tasting any food
This is why the Myrmidons named him Achilles.”
᾿Αχιλλεύς: Παρὰ τὸ ἄχος λύειν· ἰατρὸς γὰρ ἦν. ῍Η διὰ τὸ ἄχος (ὅ ἐστι λύπην) ἐπενεγκεῖν τῇ μητρὶ καὶ τοῖς ᾿Ιλιεῦσιν. ῍Η διὰ τὸ μὴ θίγειν χείλεσι χιλῆς, ὅ ἐστι τροφῆς· ὅλως γὰρ οὐ μετέσχε γάλακτος, ἀλλὰ μυελοῖς ἐλάφων ἐτράφη ὑπὸ Χείρωνος. ῞Οτι ὑπὸ Μυρμιδόνων ἐκλήθη, καθά φησιν Εὐφορίων,
᾿Ες Φθίην χιλοῖο κατήϊε πάμπαν ἄπαστος.
τοὔνεκα Μυρμιδόνες μιν ᾿Αχιλέα φημίξαντο.
I remember reading a postulated (modern, of course) etymology that suggested the name Achilles might come from the same root as Achaia. No idea if there’s any hint of acccuracy to it, but there’s a certain poetic appropriateness to the notion. (Though it would be more fitting, in that case, if his primary rival among the Trojans was Troilus, rather than Hector.)