It's a bit murky. As far as I know, it's not cleared up in any of the primary sources, and the closest is this quote from an Entertainment Weekly write-up of an interview with Abrams:
Remember how we eventually learned that “Darth” is not a first name, but a kind of title? It appears the surname “Ren” is something similar.
“He is a character who came to the name Kylo Ren when he joined a group called the Knights of Ren,” Abrams says. But that’s as far as the writer-director will go.
(Note: this interview was before film came out, hence the vagueness - I'm not aware of anything more recent)
This is being interpreted as a sign that other Knights will have names like "Brian Ren" and not names like "Kylo Stimpy", for example in this quote from cinema blend which I found in this answer to a similar question about whether his name is based on other characters' names:
We learned in an interview with Abrams in Entertainment Weekly, "Ren" is the title, which refers to a mysterious organization called The Knights of Ren. "Kylo" is the unique signifier, a name Kylo Ren chose for himself for some equally mysterious reason.
But they might be over-interpreting EW's over-interpretation of a vague statement.