The only time in The Last Jedi that Luke Skywalker uses his green lightsaber is in a flashback.
When at the end of the movie he appears on Crait as a Force projection, Luke is using the iconic blue Skywalker lightsaber – which, of course, had previously been destroyed in the film, another indication that it wasn’t actually the Jedi Master in the flesh. Earlier in the film, we saw three different iterations of what happened on the fateful night in Luke’s Jedi Temple as he confronted Ben Solo, and in two of them we see Skywalker ignite his green lightsaber. To Ben Solo, this appeared to be an act of aggression from his uncle. To Luke Skywalker, this was a brief momentary lapse in reason as he momentarily considered killing his nephew before immediately realizing his mistake.
But that’s the last time in-universe that we have seen Luke’s green lightsaber, which he constructed right before the events of Return of the Jedi. What happened to it after Luke went into exile? That question has remained unanswered, but The Last Jedi novel may give us the answer.
It’s important to note that the canonicity of movie novelizations can be a bit tricky, but generally it goes like this: if events are depicted slightly different ways in the film and in the novel, it’s the film version that is canon. But if there’s something in the novel that isn’t in the film (like, say, Han’s funeral), then it’s considered to be canon. But remember that this can be a bit tricky, like for instance Rey meeting Poe: in The Force Awakens novel the two meet near the end, but then The Last Jedi shows their implied first meeting at the end (and it goes really similarly to in the TFA novel). Generally, however, we can trust that TLJ novel is canon so long as it doesn’t contradict the movie, and this is given further credibility from the fact that Rian Johnson worked with author Jason Fry on some of these scenes and in giving information.
Anyway, at the end of the novel, right after Leia and Rey’s conversation on the Millennium Falcon fleeing Crait and right before the epilogue with Temiri Blagg, there’s a cutaway interlude back to Ahch-To, focusing this time on the caretakers, the Lanais. The novel discusses Alcida-Auka, the head of the caretakers, and the inconsistent arrival of Outsiders. And then there was the most recent Outsider, who was different. Fry writes:
The latest Outsider had been a curious one. He had arrived bearing artifacts, some of which the Lanais’s songs recalled as having been taken from the island long before. Rather than holding himself apart, he had learned the Lanais’s language and ways, appearing each month at the Festival of Return. And he had insisted on doing his own chores, gathering his own food, and performing repairs alongside them.
Luke had become a part of the village and well-received by the caretakers, unlike Outsiders before him (and unlike his Outsider apprentice, too). But Luke had disappeared, simply vanishing and leaving his robes behind. So it was up to the caretakers to clean up after the Jedi, just like it was their job to mend to the steps and to the destroyed hut and to the burned uneti tree. So, as Fry writes:
Alcida-Auka verified that one of the daughters had cleaned the Outsider’s robes and put them away in the storage hut, along with his woolens, packs, and boots. She directed another daughter to take his weapon, his star compass, and his strange other gear to the repository, where it would join other items gathered over the generations.
Did you catch that? The caretakers put Luke’s things into storage, including his star compass (which he retrieved from Pillio in the Battlefront II campaign)… and his weapon!
That’s the only indication we have of it, and it’s treated very casually. The focus here isn’t on Luke’s weapon. But it’s incredibly interesting. Now, it’s possible that this weapon is simply referring to Luke’s staff that he carries around. But it seems incredibly likely that this weapon is actually Luke’s green lightsaber. And it makes plenty of sense that he’d bring it with him to Ahch-To, as he didn’t ever intend on leaving and as he intended to let the Jedi Order die. He didn’t need to give it to someone else for safekeeping, and he didn’t need to leave it behind for the next generation of Jedi. So it makes sense that Luke would bring his lightsaber with him, keeping it among the other Jedi artifacts that he had collected.
Assuming that’s the case, then the tale of Luke Skywalker’s lightsaber is really rather simple. There’s no elaborate secret; Luke simply kept the saber with him in exile, but he never had need for it again. That’s the most logical answer.
But it also creates interesting possibilities moving forward, should future Star Wars storytellers wish to pick up on it. The caretakers took Luke’s things into storage, meaning that when another Outsider would show up, the artifacts of the Jedi Master would still be on Ahch-To. So it means that Rey could always return to the island to discover secrets left behind by Skywalker – including his lightsaber. And though Rey already has a kyber crystal from the Skywalker saber that she could use to build her own lightsaber, there seems to be another kyber crystal in storage on Ahch-To should the need arise.