"Italia" is a word with a very old etymology which meant different things at different times, and gradually came to mean the entirety of the peninsula.
Many hypotheses have been put forth as to the exact origin of the word, and some point to the ancient Greek language. Indeed, one of the Ancient Greek names for Italy, Aιθαλία (Aithalìa) could derive from the root Aith- typically tied to words having to do with fire, unsurprising as the vicinity of the volcanoes Etna and Vesuvius were, for the Greeks, major points of contact with indigenous Italians (and subsequently important centers of colonization). The very active volcano at Stromboli, akin to a natural lighthouse right in the middle of the sea halfway between the greek colonies Νεάπολις (Neapolis) and Μεσσήνη (Messene) both also in the vicinity of volcanoes, certainly adds to the plausibility of the early Greek visitors naming Italy "The Land of Flame" (or at least, Italy as they understood it; meaning the southern portion of the peninsula).
Other, somewhat more grounded theories, tie the roots of the word "Italia" to the Oscan Language of central Italy. Etruscans, Romans and Greeks referred to the Samnite tribes of the Italian interior as "VITELIU", an Oscan word meaning "Young Bull." Indeed, the bull calf was an animal that the Samnites used to personify and represent themselves as a people. The symbol of the young bull was later adopted by all opponents of early Roman expansion on the peninsula during the Social Wars of the early 1st century B.C.E. (many of whom also happened to be speakers of Oscan or one of its dialects). After the Social Wars ended with the extension of Roman Citizenship to the "VITELIU," the the word in its corrupted and latinized form "Italia" would slowly come to be a shorthand for the whole of the peninsula outside of Rome south of the river Po. However, institutionally "Italy" didn't exist yet and would be governed as an extended jurisdiction of the city of Rome.
Halfway through the first century Augustus would organize the Italian peninsula in thirteen regions. The word "Italia" would not refer to a definite political entity until Diocletian, whose reforms would bring an end to the crisis of the third century. Among his many reforms, Diocletian established a province properly named "Italia" subdivided in twelve "Dioceses." But there weren't any "Italians" just yet; rather there were Civis Romanus, or Roman Citizens, who would more likely define themselves through the most generic of the three names in the Roman naming convention: their Gens.
After Odoacer usurped Romous Augustulus in 476, the Emperor Zeno in Constantinople machinated to prop up Theodoric as Rex Italiae, or King of Italy in 493. The definition "Italia" here gets complicated, as it has half the trappings of a Roman province, and half the trappings of a Roman client state. Theodoric was awarded the title of Consul and Patrician in addition to King, but had de facto as well as some debatable de jure ability to establish his own policy towards non-Romans outside the empire's borders. And we still don't have any Italians, rather we have "Romans" and "Goths" even if the distinction seems to have been more about class than ethnic background. Theodoric himself would comment, "A poor Roman plays the Goth, a rich Goth the Roman."
After Theodoric's death in 526, his daughter Amalasunta attempted to rule as regent in the place of her five-year-old son Atalaric. The arrangement didn't last, and through plotting her cousin Theodahad came to rule Italy and Amalasunta was murdered in 535. Emperor Justinian didn't hesitate to seize on the flimsy pretext to reassert direct imperial dominion over Italy, and between 535 and 554 he would wage three campaigns on the peninsula, by the end of which the word "Italia" would come to designate a Roman province once again.
Although the name of the post-Justinian polities on the Italian peninsula did not incorporate some form of the word "Italia," being instead known by a multitude of names depending on their the degree of control by either the Italo-Lombard ruling class or the Byzantine Empire, the geographic construct of "Italy" would again be re-exhumed after the Carolingian conquest of Italy. In this period and more or less consistently thereafter, the notion of "Italy" as a geographic construct was affirmed, even though Italian notions of identity were much more complex.
I wrote a longer answer about post-Carolingian identity in Italy here, although I'd be more than happy to discuss it further.