The official name of the mission is San Francisco de Asis. This was the name given the mission by Frs. Pedro Cambon and Francisco Palou, the founding missionaries.
A small stream named Dolores ran by the mission. Later city streets and buildings were constructed around the mission and the stream was covered but the same name used for a street that replaced the stream. (The mission is located at 17th and Dolores streets.)
People began to refer to the mission as Mission Dolores relatively early. They also referred to the area where the mission was located as the Mission District.
So the answer to your question is that the official name was never changed.
There is a lot of information on this mission in the visual journey we show on our website.