As best as I understand it, the founding of the Islamic Republic of Iran in 1979 was when Persia became Iran. What was the motivation for changing the name?
This seemed to have come from Persia's freedom from the Qajars and some nationalism on the part of Germany during WWII influencing the Shah's decisions. So it was originally changed in 1935 and not 1979, unless you are only referring to the Islamic Republic addition, which was done at the Ayatollah's will, more than likely. The name of the country in Persian apparently was Iran so this was a natural choice for their own name.
You can find more here: Iran Heritage site on changing name to Iran
In 1935 the Persian government requested countries with which it had diplomatic relations to call Persia Iran, which is the name of the country in Persian. This was a grievous error based on a misdirected sense of nationalism. The suggestion for the change is said to have come from the Persian ambassador to Germany, who came under the influence of the Nazis. At the time Germany was in the grip of racial fever and cultivated good relations with nations of Aryan blood. It is said that some German friends of the ambassador persuaded him that, as with the advent of Reza Shah Persia had turned a new leaf in its history and had freed itself from the pernicious influences of Britain and Russia, whose interventions in Persian affairs had practically crippled the country under the Qajars, it was only fitting that the country be called by its own name, Iran. This would not only signal a new beginning and bring home to the world the new era in Persian history, but would also signify the Aryan race of its population, as Iran is a cognate of Aryan and derived from it
It is not true. The name Iran is old enough and comes from Ayran which means the land of Aryans but Reza Shah suspected westerners' motives in using the name Persia instead of Iran and tried to change the name to Iran again.
Reza Shah had extreme nationalist ideas, and the Nazi regime cheated him and abused these feelings in the second world war. Westerners insisted in using this name to humiliate Reza Shah as he was almost their puppet and they couldn't tolerate his new nationalist ideas and attempts for independence, so they blamed him for having close relationships with the Nazi regime and wanted to remind him their own roles in increasing his power to be the king of Persia!
As an Iranian, none the westerner's attempts and Nazi plans to abuse the naming of an old big land according to their own benefits makes sense to me. To make it more clear for you, as an Iranian, my nation and I use both names without any special purpose. My nation and I do not use Persia to say our land was a part of west colonies, as it was not. Also we do not call our land Iran to say we are the best nation in the world as Aryan is only a name of a race in genes of many different nations from India to the Europe, equal to the other races. Surely this name doesn't have any relation to Nazi's bullshits.
I think one of oldest civilizations in the world doesn't need such ridiculous tricks. I only want to correct some your mistakes about it as it is a site for expert answers to expert questions about history and needs more researches and attention before asking a question or trying to answer it.