arbitrary that it defies any
To me, at least, it would make sense if the first two letters were some kind of power grid. Assuming there is a malfunction in the grid or maintenance required, it would be more efficient to announce shipwide "There's gonna be repair / maintenance work in grid 3F on Main Deck 5 between 1000 and 1300 hours tomorrow".
Cabin occupants should immediately know whether their cabin is affected or not - by simply looking at the numbers on their door sign (if they haven't memorized these already).
Essentially, this grid locator could
serve as an alternative to sections where to locate certain areas in the ship (currently, I'd say "3F" indicates the port side in the saucer).
Despite what may or may not have been intended for the other digits of the door signs these seem to indicate room numbers.
Thus room "3F 123" (the door sign of the unseen room next to the brig in "Turnabout Intruder") could be somewhere between Kirk's cabin "3F 121" in "Journey to Babel" and Spock's "3F 125" in "Amok Time".