Aryan, (airya in Old Persian) is the endonym. Peoples of Iran used this name since, well always, for as much as we can tell, as it is attested in Avesta. And Iran, by extension simply means "land of Aryans".
And as much as Iranians themselves are concerned name was never changed- Iran was called Iran and there's that.
However Ancient Greeks first contacted Iranians then they were ruling huge empire, forged by dynasty hailing from part of Iran known as Persis (Fars), hence they called them Persians. Everyone else- starting Romans- in the Western world just adopted Greek nomenclature and went with it.
In 1930's Iranian government officially send request to the League of Nations and international community at large, asking to use Iran instead of Persia.
The name of Persians still lives as endonym of the largest ethnic group and language in Iran, so it might be a bit confusing or at least unusual: the peoples are Persians, however land they live in is Iran.
Nazi Germany really have nothing to do with this, apart of making use of outdated theories, mixed with racism. As Aryan is oldest known endonym of the IE people, it was assumed that it is the name Proto-Indo-Europeans called themselves and proponents of scientific racism simply started to use the name for European race.
No reliable person says that the name change had anything to do with the Nazis. They've been calling it Iran in Iran for more than a thousand years before Hitler or the Nazis or Germany existed. Heck, the name Iran is carved in stone in Iranian archeological sites.
https://iranian.com/main/blog/iran-history-buff/edwin-black-lies-and-distorts-irans-history.html
In fact, while Iran did have good relations with Germany prior to the rise of the Nazis, the Shah of Iran told Ataturk that Hitler was a lunatic, and furthermore Iran resented Hitler's attempts to stir ethnic unrest in the neighboring Caucasus
Furthermore, Iran during WWII gave refuge to Jews and Poles, thousands of who went there and many married and stayed there. Iran even had its own "Schindler" who saved Jews from the gas chambers while the US had turned away Jewish refugees:
https://www.ushmm.org/wlc/en/article.php?ModuleId=10005267
http://www.bbc.com/news/magazine-16190541
https://www.ushmm.org/wlc/en/article.php?ModuleId=10008210