In The Beginning
When the early developments in software were taking place (somewhere between Pythagoras and Turing), software was seen as more of a kind of mathematics.
However, as it became apparent it was useful for automated data processing for governments and organizations, companies began to develop individual and proprietary systems, starting the cycle of lock-in and repeated loss of technology investment that carried on for several decades as vendors and products changed. Software could be made better, and then redistributed at almost no cost. Hmmm...
Then There Was A License
The idea of Free Open Source Software as we now know it began with the development in the early 1980's of the concept of "free software" by Richard Stallman, formalized by his Emacs General Public License in 1988, and GNU Public License (GPL) in 1989. Richard also established the GNU project and Free Software Foundation to further his vision, and wrote some great software himself.
It took several more decades, however beginning in the early 2000's it became clear to most people that the free software vision was consolidating mind-share around the world.
Then A Community
In a co-dependent thread, in the late 1990's Eric S. Raymond and others developed the term "open source" as a more business friendly term than "free software". Open Source had a more inclusive meaning, in that licenses that were not as strict about the need to pass on modifications would also qualify for the term, and they launched the Open Source Initiative to provide a central certification organization for the many kind of licenses that met the definition.
However, by 2007 Commercial Open Source Software had effectively co-opted this term, leading the community to coalesce around the term Free Open Source Software to bring the original visions of Stallman and Raymond back together.
A Common Name
The first mention of the term "free open source software" in the newsgroups was 18 March, 1998.
This wiki's founder tried to coordinate Richard Stallman and the OSI on the de-facto term: