Spoilers for The Ballad of Songbirds and Snakes!!!
At the end of TBOSAS, it’s accepted that Lucy Gray either: dies somewhere up north, eventually returns to District 12 and lives, or returns to twelve and dies due to starvation since she wouldn’t be able to play music for a living. But there’s another possibility that wouldn’t be mentioned in the book since no one knew about it (maybe Dr. Gaul, but I’m not really sure).
Lucy Gray lives in District 13.
It makes sense if you think about it. In THG, Gale leads the survivors from twelve into the forest. Guessing from Gale’s knowledge of the forest, he’s most likely going to lead them in the direction of the meadow, which is the same direction Lucy Gray goes. I have a feeling Lucy Gray was really determined to not return to District 12, and if she does decide to return, she would have covered a lot of distance.
At the time, District 13 isn’t as developed as it is by the time Mockingjay takes place. Since it has only been a decade since they’ve signed the treaty with the Capitol, they would probably place some sort of security system in place to warn them if the Capitol try to reclaim District 13. Like a just mentioned, thirteen is new, so there’s probably going to be some evidence that there is a whole population of people underground. Whatever security system they have in place, District 13 is going to know Lucy Gray is there.
I doubt that thirteen has enough resources to use only technology for security. Even in THG, they have human guards. I’m not going into more detail about how Lucy Gray ends up in the area above District 13; I think you’ve got the point already. So now she has three choices: stay in District 13, go back to District 12, or continue north. I’m ruling out the last option because just for the sake of simplicity. There are two main pros if she stays in thirteen.
First, she’ll be away from the Capitol and Panem. “‘We’ll be free of him, of District Twelve, of the Capitol, all of it. Tomorrow. At dawn’” (483). (There was another quote I was looking for that proves my point better, but I couldn’t find it. If I do, I’ll edit this.) Basically, she’s running away in the first place to get rid of the Capitol influence. This is a common trait of both District 13 and Lucy Gray.
The second reason that Lucy Gray might stay in District 13 (which I think is what happened) was that she was forced to stay. At this point, she already is aware that District 13, or is about to realize, exists. It would be a risk for the district if someone is to know about them and be able to return to Panem. To protect their secrecy, they would force or highly encourage her to stay.
With our knowledge, whatever happens to Lucy Gray in District 13 is left a mystery. Again, this is just a theory so you may not agree with what I said.
BTW, there is absolutely no way that Lucy Gray changes her name and becomes Alma Coin. I’ve already thought about it, and I can’t bear to think that Lucy Gray is President Coin. There is absolutely no way.
Edit: Coin could be her daughter.