It’s a historical oddity that the nation (kingdom, empire, whatever it was at any particular point in history) of Iran was rarely called “Iran” by anybody other than Iranians until 1935, even though most Iranians had been calling it “Iran” for millennia. The rest of the world didn’t catch on until Reza Shah Pahlavi (d. 1944) requested, in December 1934, that as of the next Iranian New Year (Nowruz), all foreign governments should henceforth stop referring to his country as “Persia” and start calling it “Iran.” Sometimes you’ll see this related by Western writers as “Reza Shah changed the name of the country from Persia to Iran,” but that’s dumb and wrong, because, again, Iran was always the name of the country. “Persia” was, for the most part, what’s known as an “exonym,” which is the term used when a group, place, language, or some other national feature is given a different name by people who aren’t part of that group, or don’t live in that place, or don’t speak that language, or all of the above.
With all due respect to Herodotus or whichever Greek writer convinced the rest of the world that the land between Mesopotamia and the Indus River was properly called “Persia,” very few people who actually lived in that place would have recognized it by that name. You can find inscriptions from the second century CE that refer to the place as “Iran,” and that word or its variants (Iranshahr, Iran-zamin) go back far earlier than that. The ethnic identifier “Aryan,” from which “Iran” derives, can be found in use as far back as the Achaemenid period (500s BCE). It can also be found in the Zoroastrian scripture, the Avesta, which reflects a tradition that goes back much earlier than that, though the earliest extant complete copy of the Avesta only dates to the 14th century.
The label “Persia” is essentially a historical error: Cyrus the Great (d. 530 BCE), who founded the Achaemenid dynasty, started out as the king of Parsa, the region of southern Iran that is today called Fars (the “p” was traded for an “f” when the Arabs conquered it because Arabic has no “p” sound). Greek writers, demonstrating that a little knowledge can be a dangerous thing, applied the word Parsa to the entirety of Cyrus’s domains, even though the actual Parsa only referred to one kingdom amid his vast empire. For some reason (which I think we can probably figure out if we really think about it) that name stuck in Western usage when referring to the eastern part of that empire, even though its western domains (which included, at one time or another, Anatolia, Egypt, and Israel) were blessed with a more nuanced treatment.
After the Arab conquests, the whole idea of referring to historical Iran by a single name, whatever that name might be, became defunct, as the former “Persian” Empire was broken up into several provinces: Fars, for example; Iraq al-Ajam (“foreign Iraq”), the region in central-western Iran bordering Mesopotamia (which was called “Arab Iraq” or Iraq al-Arab); Sistan (southeastern Iran); Khurasan (northeastern Iran, including most of modern Afghanistan); Mazandaran (the southern Caspian shore); and so on. Once the caliphate fell in 1258 and truly independent kingdoms arose once more in Iran, the concept of “Iran” as a geographical unit became relevant again. The advent of the Safavid dynasty in 1501 really marks the return of “Iran” to the world in the form we know it today, although the Safavids, during their relatively brief heyday, ruled a territory that was considerably larger than the modern nation of Iran.
If you really care about this subject and/or have a deep sense of self-loathing, you can also take up the eternal debate over whether to call the predominant Iranian language “Farsi” or “Persian.” As we’ve already seen, these two words are effectively identical from an etymological standpoint, and Farsi is the word used by speakers of the language to refer to the language itself. But there is a school of thought (with which I actually agree) that says that it’s better for foreigners to refer to the language as “Persian” because it conveys the history of the language and its people better than “Farsi,” which to a layperson means little and which could be taken to refer too specifically to that one single part of Iran, Fars. But I digress.
Why did Reza Shah ask the international community to start using “Iran” in December 1934? Apart from the fact that “Iran” was the more accurate name, he did have some geopolitical reasons, and unfortunately they involve the Nazis. As “Aryan” fever gripped Germany, the Nazis convinced the shah, with whom they had good relations, to emphasize the name “Iran,” which as you saw above derives from the same Old Iranian word whence we get “Aryan.” This was supposed to affirm some sort of historical bond between the Germans and the Iranians, and lend historical depth to the Nazis’ idea of a pure “Aryan” race. Reza Shah, who appreciated the Nazis not so much for their racial science as for their anti-Britishness, was happy to go along.
Reza’s request also may have had something to do with the fact that he had a legitimate claim to being the first “Iranian” ruler of “Iran” since the Arab conquests–the Safavids, who get credit for restoring “Iran” as a political entity, were at least part-Kurd, and their authority rested heavily on the strength of their Turkic army. The succeeding Afsharid and Qajar dynasties were Turkic by ethnicity (the Afshars and Qajars were two of the largest tribes in that Safavid Turkic army). Maybe Reza saw the reassertion of Iran’s actual name as a way to aggrandize himself as a true “Iranian” shah.
Today there’s an interesting phenomenon whereby some Iranians in the diaspora often prefer “Persia,” which (to them at least) conveys the historical weight of those great empires of antiquity, over “Iran,” which (again, to them) conveys the most unflattering impressions of the Islamic Republic. The dual name thing also occasionally gives rise to some of the silliest of the generally silly mainstream American media takes on Iran, like this thing from Cliff Irving at The Daily Beast called “Are We Negotiating with Iran or Persia?” One of those is supposed to be bad, I guess, but I haven’t actually read the thing and I am completely unable to make myself care what his point is, assuming he has one. But all your hot Persia takes aside, Iranian history is pretty unambiguous that “Iran” has always been the proper name for, well, Iran.