Internally they never did. Within Persia the name has pretty much always been a variant of Iran. At least as early as the Sassanids, they called it Iran or Iranshahr. I'm not sure what the Achaemenids called it, if they did at all, but since them it's been Iran. Exact spellings and pronunciations varied over the millennia, but it's consistently remained Iran or a close variant.
Persia is the Greek name for the Achaemenid empire. Probably named after Cyrus the Great's tribe, the Pars. Once coined the name remained the name for the people and region in Greek, and afterwards within the Roman Republic and Empire. When the Empire faded, its successor states continued to use the Persian name.
In 1935 Reza Shah Pahlavi requested that foreign governments use the traditional local name rather than the foreign name for the country. Internally this wasn't a name change. It was only a change within in the foreign ministries of the world.
Not discouraging new answers, this was answered here: https://www.reddit.com/r/AskHistorians/comments/236u3r/why_did_the_people_and_leaders_of_iran_not_name/