Last year, while working on Empowered Judaism, Elie Kaunfer emailed me and asked if I knew the origin of the term "independent minyan", and said that he had it in his head that I had coined it. My first reaction was complete disbelief - of course I didn't coin "independent minyan"! Whenever I started using the term, I was just using a term that was already out there. My second reaction was that I was unable to find evidence to disprove the claim that I coined it! (Isaac Asimov coined the word "robotics" based on the existing word "robot", but not on purpose -- he thought it was already a word.) I searched through my old email, and the earliest instance I could find was an email I sent on December 30, 2002 to MAIL-HAVURAH (the predecessor to the current NHC email lists), which began "I have been involved in the creation of Kol Zimrah, a new independent minyan in New York City (on the Upper West Side, so far)." (The rest of the email went on to describe and promote Kol Zimrah, and ask questions about how to find davening spaces in New York and how to get people involved in the community.) In Kol Zimrah's next announcement to its email list (sent on January 15, 2003, announcing its January 24 service), it described itself as "NYC's newest independent minyan".
In late January 2003, when Kol Zimrah set up its first website, it included a "Directory of Independent Minyanim in NYC", to keep track of all these minyanim that were sprouting up like weeds. It was an exciting time: at least 8 independent minyanim had started in NYC in 2001 and 2002 alone (4 of which still exist today). And I'll clarify here, since (whether or not I coined the term) it was my idea to start this directory (according to an email to the other KZ organizers dated January 30, 2003), so I have some idea of the original intent. "Independent minyanim", in regard to this directory, was understood in contradistinction to synagogues, or to non-independent (e.g., synagogue-affiliated) minyanim, not in contradistinction to "havurot". The directory didn't include any first-wave havurot, not because we didn't consider them to be "minyanim", but because, as far as I know, none (that qualified as "independent") existed in NYC at that time: the New York Havurah had long since gone the way of Radcliffe College (that is, I hear they still have reunions, but otherwise no longer function actively), and the West Side Minyan and its spinoff Minyan M'at had become part of Congregation Ansche Chesed (and were thus not "independent"). Likewise, the directory was not restricted to minyanim founded after a particular date, though most of the independent minyanim that existed in NYC at that time had in fact been founded after 2000 (the only exception, I believe, was KOE, which was not part of the new tidal wave of minyanim, but not generally considered part of the "havurah movement" either).
Kehilat Hadar's mission statement on its website describes itself as "an independent, egalitarian community committed to spirited traditional prayer, study and social action", so this may have gotten "independent" out there in regard to these communities, though not the exact phrase "independent minyan", since Hadar doesn't officially describe itself as a "minyan" (though its email address was egalminyan at hotmail dot com from before it was called Hadar until not so long ago, making them the last people in the world still using Hotmail). This is explained in Empowered Judaism: "We intentionally did not name the community Minyan Hadar, because we aspired to be something more than just a minyan."
Certain that I couldn't possibly have coined "independent minyan", I asked Facebook if they knew of any uses prior to December 2002, and sure enough, people with LexisNexis access came up with some earlier citations. The first was from the Philadelphia Inquirer on November 17, 1988, in an obituary for one Gerald Margulies: "Mr. Margulies also devoted time to being a leader of an independent minyan, or prayer group, that gathered at the Y.M.H.A. on City Avenue, and was a beloved uncle to his 15 grandnieces and grandnephews, family members said." (Does anyone know anything about this minyan? The Philly people I asked had never heard of it.)
The second was from an article about the Jewish community of Prague in the Jerusalem Report, August 24, 1995. Discussing people who are considered Jewish by Israel's Law of Return but do not have Jewish mothers, the article said: "These Jews are welcome in some of the Jewish cultural organizations and can even find a home in an independent minyan sponsored by the Reform movement called Beit Simcha." Of course, if it's sponsored by the Reform movement, then it's not considered "independent" the way we would generally use that term today. Based on the context, the apparent meaning is that it was independent from the official Jewish community institutions (not a concept that exists in the United States). The article goes on: "While synagogue buildings stand empty or underutilized, Beit Simcha pays rent for a basement room in an apartment building. The atmosphere is reminiscent of an American havurah-style minyan: comfortable chairs arranged in a circle, Zionist posters on the wall."
Still, I haven't found any instances of "independent minyan" in reference to the 21st-century phenomenon earlier than December 2002. The early articles on this wave of minyanim used other terms to describe them. For example, both the August 10, 2001 article in the Forward and the August 2, 2002 article in the Jewish Week referred simply to "new minyans". The former also called the phenomenon a "new havurah movement" (italics in original), and called the minyan that would later be named Hadar a "nondenominational minyan" and a "fortnightly minyan". The first 21st-century use of "independent minyan" in the media might be Jay Michaelson's article in the November 14, 2003 Forward, "A Prayer Group of Their Own: Kol Zimrah and Other Do-It-Yourself Minyans Unite the Independent-Minded", which uses this term repeatedly. As the title suggests, Kol Zimrah is featured prominently in this article.
So the evidence seems to indicate that Kol Zimrah was instrumental (as it were) in establishing the term "independent minyanim", completely by accident. But it's possible that there's still more evidence to be gathered. Do you remember when you started using the term "independent minyan", and where you first heard it? Do you have email correspondence using this term earlier than December 30, 2002?
UPDATE (October 2013): Mark Berch sends us the article "Understanding Synagogue Affiliation" by Steven Huberman, Ph.D., published in the Journal of Jewish Communal Service in 1985. The relevant paragraph states:
Los Angeles has a variegated network of synagogues. There are synagogues of every ideology—ranking from large, old, well-established temples to traditional Chassidic Shtiblach. There are independent minyanim, which serve as alternatives for those seeking religious expression outside of conventional institutions.
From the context, I think it is reasonable to conclude that "independent minyanim" is used here in the same sense that it is used today. So there's an answer.