The word "Italia" dates back to the pre-Roman period.
Many hypotheses have been put forth as to the exact origin of the word, and some of these point to Greek. The Ancient Greek word Aιθαλία (Aithalìa) could derive from the root Aith- linked to words having to do with fire, unsurprising as the Italian volcanoes Etna and Vesuvius marked major points of contact between Greeks and Italians (and later become important centers of colonization). The very active volcano at Stromboli could have been akin to a natural lighthouse right in the middle of the sea between the greek colonies Νεάπολις (Neapolis) and Μεσσήνη (Messene) which were also within sight of volcanoes. It's certainly plausible that early Greek mariners called Italy "The Land of Flame."
Other theories tie 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, he Samnites used Bulls and Bovine imagery to represent themselves. The symbol of the young bull was later adopted by all opponents of Roman hegemony on the peninsula during the Social Wars of the early 1st century B.C.E. (who were speakers of Oscan or one of its dialects in any case). After the Social Wars ended with the extension of Roman Citizenship to the "VITELIU," the the word in its corrupted form "Italia" would slowly come to be a shorthand for the central and southern part of the peninsula (the northern part was called "Cisalpine Gaul"). 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 organized the Italian Peninsula in thirteen regions, but the word "Italia" would still not refer to a definite political entity until Diocletian, who in the reforms that would bring an end to the crisis of the third century established the province of Italia subdivided in twelve "Dioceses." As for "Italians," there weren't any as of 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 and their associated origin story.
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 has the title of Consul and Patrician in addition to King, but has de facto as well as some debatable de jure ability to establish his own foreign policy. 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 astutely 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 machinations 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 imperial dominion over Italy, and between 535 and 554 he would wage three campaigns on the peninsula, by the end of which "Italia" would be a Roman province once again.
The reestablished roman province of "Italia" would not last long. The Empire's resources were stretched, and with the majority of troops recalled to fight against the Persians in the east, the last great migration, the Lombard descent into Italy, could take place with supreme ease in 568. The word "Italy" starts to fall off the map. In the long struggle between Lombards and Empire, the peninsula was divided into multiple parts: "Longobardia Major," or Northern Italy ruled by the King of Lombardy (de jure from Mediolanum but de facto from Pavia); the "Romagna" or coastal area held by Byzantium around the delta of the River Po ruled by the Exarch in Ravenna; the Duchy of Rome, in theory ruled by the Roman Senate but in practice dysfunctional without intervention of the Exarch; and Longobaria Minor, ruled by Lombard Dukes in the South of modern Italy.
A near-constant state of conflict existed between the Lombards and the Byzantines. The Lombard King Desiderious (d. 786) got very close to finally kicking the Byzantines out of Italy: A successful offensive saw the encirclement of the Romagna, allowing Desiderious to march south against Rome unopposed. Seeing the Byzantine army pinned down and fearing a loss of independence for the Bishop of Rome (who had been pretty much given a free hand to rule the modern Lazio Region), Pope Adrian excommunicated Desiderius and asked Charlemagne to come to his aid. Charlemagne, even though he was busy conquering what would later become Germany, was immediately flattered and marched for Italy.
After Charlemagne's conquests the term "Regnum Italiae" emerged once again in 781, to denote the conquered Lombard lands which Charlemagne gave to his third son, Pepin. The "Kingdom of Italy" didn't refer to the entire peninsula though, as southern Italy had never been conquered and was still ruled by Lombard dukes. "Italia" refers to the northern portion of the peninsula, while the southern part was "Longobardia". This setup or North/Central Italy being referred to as "Italia" and southern Italy being referred to as something else would continue until Italy was completely unified in 1861.
After Pepin's untimely death and the wars of succession following Charlemagne's death, the Kingdom of Italy was given to Lothair (a grandson of Charlemagne) in 795. The uneasy but relatively stable transitions between emperors in the "Crownsharing" deal under the Carolingan dynasty ended in 887, after the deposition of Charles the Fat. Charles was the last Emperor to simultaneously hold the title of King of Italy, France, and Germany, and had even been elevated to the Imperial Throne from the position of King of Italy. However, Charles possessed no direct heirs. So the fight between innumerable female-line descendants was on, and in Italy these were Berengar, the Marquis of Friuli, and Guy, the Duke of Spoleto. Partly because of the distance separating Friuli and Spoleto no pretender to the crown was able to assert himself. An incursion by Arnulf, who in the meantime got himself proclaimed King of France and Germany (but not Italy) in 894 didn't help the situation either
After the resources of the Italian lords were spent, minor nobles backed the nearby Kings of Burgundy and Provence (two short-lived post-Carolingian breakaway states) who contested the title until Berengar, the Marquis of Ivrea and grandson of the deposed Berengar of Friuli, machinated the support of Emperor Otto and got himself crowned King Berengar II. Until Otto decided to take the crown for himself, that is.
The Ottonian Emperors Otto I, Otto II, Otto III, and Henry II all travelled to (North/Central) Italy to be crowned King of Italy by an Italian Bishop (if he was the Bishop of Rome or Milan was of little consequence at this point). Interestingly, the Ottonians came to rely on Italian revenues to contrast the unruly vassals in Flanders, Bavaria and Luxembourg. Until, alas, Henry II died without issue in 1024. And here it gets interesting, because although Conrad was elected emperor with minimum fuss in Germany, in Italy his ascension was not entirely recognized.