I didn't coin the term. I saw it somewhere on Tumblr (where else?), perhaps at Be a SJ Ally, not a SJ Sally, and verified that it was becoming common at Urban Dictionary: social justice warrior.
I get credit for it because this blog's older entries were imported from elsewhere, so the site looks older than it is. And I began using the term in the science fiction community, where SJWs were very active but no one had identified them, so it's fair to say I helped popularize it.
I don't like the term because it's so open to misunderstanding. I would like something more precise, like Bullies Who Appropriated the Good Name of Social Justice to Excuse Verbal Abuse, Death Threats, Doxxing, and Censorship, but people like short names, and I can't offer an alternative to SJW that's as terse. Most of the alternatives—clicktivists, keyboard warriors, etc.—leave out the context, and for SJWs, "social justice" is the crucial concept. Nevermind the fact that traditional social justice workers like Dorothy Day and Dom Hélder Pessoa Câmara are tweeting from heaven under the tag #NotYourShield.