In The Force Awakens, Finn indicates that stormtroopers are kidnapped and indoctrinated at a young age. This would imply that they have family-assigned names before joining the First Order.
Do any official sources specify Finn's birth name?
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JJ Abrams already knows, but he isn't talking:
I will only say about that that it is completely intentional that [Finn and Rey's] last names aren’t public record.
John Boyega [Finn] has also said:
“I didn’t know much going in... I just remember during my time screen-testing, I was like, to Daisy [Ridley], ‘there’s no way that our stories are so simple,’ and we still don’t know! So I’ve still got some conspiracy theories as a fan as to where Finn comes from and I’m still trying to figure that out, but I like that it’s a mystery.”
On a side note, the numeral portion of Finn's identification number as a stormtrooper is significant in itself. His ID is "FN-2187"; "2187" was the number of the cell in which Leia was detained aboard the Death Star in Episode IV. Even that appearance of the number "2187" was a reference to something else: as a film student at the University of Southern California (USC) Cinema School, George Lucas saw a short film called 21-87, which was a huge influence on him and reportedly inspired him to make his debut film THX-1138 (which is set on the date 5/14/2187) and he inserts references to it (and to THX-1138) in many of his movies.
In fact, the concept of the Force was also inspired by 21-87:
One of the audio sources Lipsett sampled for 21-87 was a conversation between artificial intelligence pioneer Warren S. McCulloch and Roman Kroitor, a cinematographer who went on to develop Imax. In the face of McCulloch's arguments that living beings are nothing but highly complex machines, Kroitor insists that there is something more: "Many people feel that in the contemplation of nature and in communication with other living things, they become aware of some kind of force, or something, behind this apparent mask which we see in front of us, and they call it God."
When asked if this was the source of "the Force," Lucas confirms that his use of the term in Star Wars was "an echo of that phrase in 21-87."