Accordingly to this source:
During preproduction, Abrams was reminded of Phantasm, a 1979 horror film that featured a gaunt, terrifying figure known as The Tall Man and a flying, silver sphere that bores into its victims’ bodies like a bullet crossed with a drill-tip.
“Phasma I named because of the amazing chrome design that came from Michael Kaplan’s wardrobe team. It reminded me of the ball in Phantasm, and I just thought, Phasma sounds really cool,” Abrams says [...]
Ok, but in the context of the movie, what does Phasma means? As DVK pointed in his answer, this is probably simply her true name. Also, as DVK already pointed in his answer, she is one of the highest possible officers in the First Order. So, she is not just another random generic stormtrooper that nobody really cares about the name or that simply receive generated codes like FN-2187 (which is Finn's code).