There was a huge social revolution in Iran at that time (1979-1980) and the government was changed from monarchy to Islamic Republic. As a result, and as part of the "Cultural Revolution", many aspects of the Iranian culture were changed in order to reflect the new values of the society.
Before the revolution Esteghlal was called "Taj" which means "Crown", a symbol of monarchy. They renamed the club from "Taj" to "Esteghlal" which means "Independence".
There was an attempt to change the name of Persepolis as well. Perspolis is the name of an old Persian city. They tried changing the name to "Pirouzi" which means "Victory". However since Perspolis did not symbolize the Shah (Iran's King) or his government in any way, the old name of Perspolis remained and many use both names for the club nowadays.