Persia was the name used by the Greeks, Romans, and those who came after them. It specifically refers to the province of Pars (Fars) which was the cultural and political center of the country. Persians, however, have always referred to their own country as Iran which comes from aryānam, which means "[the land] of the Aryans." Therefore changing the name would make sense since it evokes the long and rich history of Persian civilization.

Source: Bernard Lewis, The Middle East: A Brief History of the Last 2000 Years.

One specific reason for the change was geopolitics. Reza Shah, the ruler of Iran in 1935, asked world leaders to change the way they called Iran as a pro-Nazi gesture despite the fact that the Nazi idea of "Aryan" was different from the historical ethnicity. This was due to the fact that Germany had built up a considerable relationship with the oil-producing country and oil was a vital strategic resource leading into and during World War 2.

Source: Abbas Milani, The Shah.

Edited for a bit of context.