Short answer: we don't know. The name of the figure "Italos" (Ἰταλός) has an old pedigree in Greek writing, as far back as Thucydides, who just calls him a "king of the Sicels") (6.2.4). The actual word Italia almost certainly came before this, and the myth(s) was built around the word. As far as myths go, it is not well known now, and there are multiple different versions with wildly-varying narratives, but "Italos" is mentioned a decent amount in a wide variety of ancient sources. It would make sense that the word originates among those dwelling on the Italian peninsula, but written evidence of the word in Greek (Ἰταλία) is much older than the oldest evidence of the word written in any actual Italic language. It is certainly possible that a Greek term for the peninsula, based on some obscure and long-lost myth, wound up being the accepted universal term even in Latin. That would be deliciously ironic, given that the Latin term for the Greek peninsula, Graecia, became the default term vs the original Greek term Hellas (Ἑλλάς).

We have no idea what the etymology of the actual word is, whatever its origin. It has no obvious Indo-European cousins except for the enigmatic term in Oscan (an Italic sister-language to Latin) "viteliu / vitelliu", which was what they called the Italian peninsula (or a smaller portion of it in the southwest). Based on that, it might be related to an Italic root that means "calf," as Dionysius of Halicarnassus reported (Roman Antiquities 1.35). Ernout (Dictionnaire etymologique de la langue Latine) thinks it is just a silly pun ("n'est qu'un calembour"). He tentatively suggests an indigenous (i.e., not Indo-European) origin, possibly Illyrian.