According to a 06 August 2009 article by Muhammad Sahimi:
In another speech, when he [ Bazargan ] mentioned the Ayatollah's name, the crowd chanted a Quarnic verse that praises the Prophet, his family and descendants. This is a tradition in Iran to chant this when the Prophet is mentioned. But, not only did the crowd chant the verse for the Ayatollah, but they did so three times, whereas it is chanted only once for the Prophet. Bazargan got angry and said, "Come on! This is done only once for the Prophet, but you do it three times for the Imam? What is going on?"
According to Bi Parchamaan, The Flagless Ones (2011):
The crowd begins to show its respect to Khomeini by saying in chorus, 'peace be upon Mohammad and his family'.
Praising the prophet Mohammad and his family whenever the prophet's name is mentioned is a religious tradition, people nearly always wing a word of praise heavenward on hearing his name. Beginning with the revolution, we were supposed to show the same respect to Khomeini. Whenever somebody said 'Imam Khomeini', we were coached to praise the leader in chorus. This is what we were doing every time we heard the imam's name, but this threatening son of our local leader shouts, “I don't understand. You praised the prophet Mohammad once and praised our very own imam only once as well?” For a while, we don't say anything and then suddenly break out praising Khomeini continuously for three repetitions. I have no idea why we choose three rather than two or four or some other number.
From this night on, we praise Khomeini three times whenever somebody calls him by his name during daily speeches.
Also, Women's Islam: Religious Practice Among Women in Today's Iran (2013) says:
... the idea of offering salutation three times as a tribute to Imām Khomeini, and only once to the Prophet Muhammad, is not only uncommon, but is openly objected to by those with oppositional political views. The latter argued that it was an act of blasphemy (kofr/shirk) to decrease the salutation of the Prophet to one and instead offer it three times to servants of God like Imām Khomeini.
So this is the practice that started in 1979 in Iran, but it is not accepted by all Shia.