UPDATE2: For those who have concerns about novelization being canon yet way different, the WGA film script confirms that other stormtroopers have different designation prefixes:
Don't move! TK-338, we have targets in custody!
UPDATE: it seems to just be a "corps" number.
It was the start of FN-2187’s own streak, however. The next trainee to go at him was from a different cadre, the FO group, also armed with a mace and shield. The fight lasted three seconds. FN-2187 feinted an overhead blow with the mace, and when his opponent brought his shield up to parry it, he hit him instead with his own shield and knocked him flat. His next two opponents were also FO designations, another with a force pike and one with a shield and sword. The second of those took the longest, almost a full minute, before FN-2187 managed to knock away his opponent’s shield, and then it was a simple matter of waiting for an opening and striking at the right time.
("Before the Awakening" prequel novel by Greg Rucka)
The next combatant was from another training cadre, FL, but FN-2187 didn’t really notice and didn’t much care.
“Fresh meat,” one of the stormtroopers said. “Who’s who?”
Slip grinned and indicated himself, then the others. “FN Corps. Slip, Zeroes, Nines, and FN-2187.”
As user @bmwurm pointed out wisely on a comment in a linked question, the known designations seem to possibly indicate absolutely banal numerical progression, using base 26: "FN" is followed by "FO" followed by "FP"....
I'm not aware of canon explanation at the moment, but I would like to disprove a popular theory:
FN does not stand for "Finalizer", a Resurgent-Class Star Destroyer we see in TFA.
In SW: Before the Awakening (TFA prequel YA book), we see them having this designation well before being deployed, from the start of the book.
THERE WERE four of them in the fire-team, and because shouting out things like, “FN-twenty-one eighty-seven, watch your back!” was a mouthful, especially when the blaster fire was searing the air around them, they’d defaulted to shorter versions. In front of the officers, in front of Captain Phasma especially, they always used their appropriate designations, of course. But in the barracks and in combat, they used the names they’d given one another or the names they’d given themselves.
FN-2199, he was Nines, because he liked the sound of it, simple as that. FN-2000 had told everyone to call him Zeroes, because he was proud of the fact that he’d landed such a straightforward number as his designation. He thought it made him special, and either nobody had ever told him that being a “zero” wasn’t exactly something to be proud of or he didn’t care.
However, they only ship out to Finalizer later in the story:
hey moved from the base to a transport and from the transport to orbit, traveling with another half dozen of the trainee squads, all of them in their armor and with their rifles. The rifles were new, no longer the training version but the real thing, F-11D blaster rifles, loaded with live ammunition and fully charged for battle. Their first look at the Star Destroyer, majestic and ominous at once, was through the hull windows as it came into view—almost impossibly small at first, then growing to become almost impossibly large as their shuttle sped toward it.
“This is for real,” Nines said, and FN-2187 thought there was awe in his voice, as if he’d never expected them to make it that far.
“Did the captain say where we’re going?” Slip asked. “What we’re doing?”