I think a fairly common misconception is that, because Iran is the most populous Shia country, it is also the origin Shi'ism.

In fact, Shi'a Islam developed largely in a Arab context. Najaf and Karbala, two of the holiest cities within the Shia faith, were (and are) thoroughly Arab. Arab-dominated Iraq; Jabal Amel in Lebanon; Eastern Arabia, particularly Bahrain: these were three of the major theological centres of the sect from its development until the 16th century.

Until the 16th century, Persia was majority Sunni. In 1501, Shah Ismail established the Safavid dynasty over Iran. He was a Shia Muslim and established this as the state religion. While there had been Shia-ruled states in the past (notably, the Qarmatian republic in east Arabia and the Fatimid Caliphate in Egypt, both in the 10th century), It's fair to say that this was the most successful Shia state yet (the Fatimids never attempted to force the Shia faith onto their ruled people, and the Qarmatian movement fizzled and devolved into local emirates, though some of these did retain the Shia faith).

Since Shia Islam had no strong foundations within Iran, the Safavids had to import their clergy from outside. Lebanese and Bahraini Shia were two of the most significant centres from which theologians were imported. Over time, Iran developed its own centres of religious learning (notably Qom). Ultimately however, the country owed its successful conversion to Shia Islam to Arab Shia.

See Juan Cole, Sacred Space and Holy War, 2002.