It seems Stormtroopers from The First Order only have some sort of designation instead of a name. However, Captain Phasma seems to break that rule.
As far as I know, we don't see any other captains in the First Order so I can't tell if this is common, but who named Phasma, and why?
I'm still curious, as I'd imagine you're promoted from trooper (who has a designation) to Captain (who has a name).
This is discussed in the 2017 novel Phasma. The novel tells us a few things:
Receiving a name (and also distinctive armour) seems to come with the territory of being a stormtrooper captain, and is at the pleasure of high-ranking members of the First Order (all captains we know of were promoted to the position by General Brendol Hux, one of the most senior First Order commanders):
"How would you like to have a name other than a number? Captain Cardinal has quite a ring to it, does it not?"
It was an unusual honour, and Cardinal's heart had swelled.
Phasma Chapter 34
Promotion to captain is an extremely rare honour, and is the highest rank in the stormtrooper corps; by the start of the novel there are only two captains, and they split the task of training stormtroopers (though, as noted elsewhere on the site, Phasma's rank doesn't really reflect her importance within the First Order):
Out of all the identical troopers, one had risen above. One had flown higher. Cardinal was to be more than just a stormtrooper - the highest of the stormtroopers. Hence the color, hence the unique name. Cardinal for the red bird, Cardinal for his standing as the first of his kind, the first trooper with a name and with red armor.
And then Phasma showed up.
She never had to give up her name for a number, as Cardinal had. She didn't get red armor, thank goodness, but it was obvious that Brendol considered her special, too. Given a captain's cape early in her career, the only one other than Cardinal. Offered the chance to mold the training program to her specifications, to program the sims and find new ways to challenge the teens who'd graduated from Cardinal's own curriculum.
Phasma Chapter 34
Theoretically, a captin's name can come from anywhere; Cardinal's birth name is established as Archex, he's given a number upon joining the First Order, and his captain's name is chosen at a whim by Hux:
"Ah, CD-0922. So upright in your red uniform. That's why you're first among my men. Cardinal, even" [Brendol Hux] laughed at his own joke.
Phasma Chapter 34
Despite that, Phasma's name is the one she was born with1:
"Cardinal, this is Phasma," Brendol had said.
[...]
"Has she been assigned a number, sir?" he asked, hating the squeak of desperation in his voice.
Brendol smiled that secret smile of his and chuckled. "Oh, no. Like you, she is a special case.
Phasma Chapter 11
The name was evidently chosen by Phasma's mother, who we never meet, and we're not told what it means in the language of Phasma's people. Why she was permitted to keep her own name is also not mentioned explicitly, but it was likely part of the extended deal she worked out with Brendol Hux when she joined the First Order2
1 Not really, because Phasma's people have an extremely high infant mortality rate and don't give names to newborns, but it's close enough for our purposes.
2 Which, in the interests of not spoiling the bulk of the novel, I'm not going to describe in more detail. If you want to learn more, go read Phasma; it's good
This is not 100% certain in canon, but from Visual Dictionary, we get the impression that:
Phasma isn't just "one of stormtrooper captains" - she may have the name of that rank, but her official position is significantly higher, and basically equivalent to a General (or more likely, a bird Colonel - she still seems formally subordinate to both Hux and even Kylo Ren - she asks Ren for his orders and calls him "Sir" on Jakku, in the novelization).
Charged with commanding the stormtrooper forces of the First Order, Phasma's true rank is higher than the simplistic label "captain" would suggest. Although her position could easily afford her a well-appointed war room far from the battlefield, Phasma insists on seeing combat operations firsthand, and shuns any comfortable trapping of elevated rank.
She's high enough to be nearly equal to General Hux:
Phasma serves as the third partner in the unofficial command triumvirate of the Starkiller operation, alongside Kylo and Hux
... Including being able to disagree with him
Phasma disagrees with General Hux over what it takes to be a soldier
There are also some hints that she wasn't "promoted from a serialized trooper" (insert your programming "marshalling" jokes here):
She spends her every waking hour honing her combat abilities... qualified expert in all FO small arms...
Now, by itself, this quote doesn't say much, BUT, to me, someone who cam up the ranks of stormtroopers would kind of be expected to have honed abilities and be arms expert - that's generally what you get when you promote best combat troopers. Whereas, if she's an officer who's not from stormtrooper ranks (like Hux), such combat skills are unusual, and worth emphasizing - and as the quote shows, emphasized they are.
All this leads me to a conclusion that she's a senior command officer and as would be normal in such positions in Empire, that's her actual true name. The First Order likely only gave numeric designations to line troopers (a holdover from the good old days of the GAR clone serial numbers) - not to the brass.
Note that this is different in the Empire/First Order, from the GAR, where even captains had serial numbers - e.g. infamous Captain Rex commanding 501st was actually CT-7567
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).
