Star Wars gave Luke Skywalker a new, green lightsaber in Return of the Jedi, but the movie failed to explain just when and how he made the weapon.
Luke Skywalker debuted a new, green lightsaber in Return of the Jedi, but just when in Star Wars canon was the weapon built, and how? When it comes to Luke's Jedi weapons, it's the blue Skywalker lightsaber that has received most of the attention. It's this blade he wielded in the first two original trilogy movies, that's used by Anakin in the prequels, and then is passed on to Rey in the sequels. Meanwhile, Luke's green lightsaber is missing in action, with its only appearance in the Disney movies during a flashback, and just a hint in The Last Jedi's novelization that it was with him on Ahch-To.
However, that shouldn't detract from the importance of Luke's green lightsaber, which was a big deal when Return of the Jedi was released in 1983. Up until then, audiences had only seen either blue (Luke and Obi-Wan) or red (Darth Vader) lightsabers, so introducing green into the mix was a real change. Luke needed a new lightsaber after losing his hand during his duel with Vader on Bespin, and so he eventually had to set about making himself one, which also helped cement his becoming a Jedi.
Although there's a one-year time gap between The Empire Strikes Back and Return of the Jedi, Luke built his green lightsaber more towards the events of the latter. A deleted scene from Return of the Jedi shows Luke putting the finishing touches on his new weapon in a cave on Tatooine, with R2-D2 and C-3PO there with him. They head off to Jabba's Palace, allowing Luke to make his own rescue attempt a little later. The scene was likely scrapped to preserve the surprise, but the moment itself is better fleshed out in the junior novelization of the movie, Return of the Jedi: Beware the Power of the Dark Side, from 2015. Set in 4ABY, the same year as the movie, the canon book delves a little more into the construction of Luke's green lightsaber, revealing that completed it in a cave on Tatooine, using instructions found in Obi-Wan's journal in his deserted hut:
"And yet he seemed to know exactly what he needed to know. Reaching out with his mind, he found the right pieces - some easily purchased, some much harder to acquire. After leaving Obi-Wan’s hut, he knew he had at last collected everything he needed. While our other heroes bustled about preparing for the rescue, Luke retreated into the solitude of a desert cave and puzzled over the pieces… In the end it took not just physical tools but also the Force to put it all together and bring the crystal inside to life."
There isn't much more in terms of detail offered, but the canon explanation does broadly keep with the bulk of the Star Wars Legends one, which was explained in Shadow of the Empire. There, however, the process was far more detailed, with Luke writing his own account of the steps he took to build the saber, including collecting most of the materials from around Mos Eisley, and making his own synthetic kyber crystal since they were in scarce supply. It took him over a month to complete, despite the book saying that "a Jedi Master in a hurry could construct a new lightsaber in a couple of days", which shows even then he wasn't at that level yet. While that isn't canon, it seems at least some of the actual building process was similar to the old EU. One interesting point of question is just where Luke got the kyber crystal for his lightsaber - rather than it being synthetic, one fun theory suggests it's actually that of Obi-Wan's master, Qui-Gon Jinn, though it's rather unsubstantiated. Still, Luke building his lightsaber in Obi-Wan's hut from his schematics makes sense, and serves as a useful explanation for why the weapons look so alike.
Of course, because Star Wars canon is messy - even in the Disney era, despite them trying to streamline it - the novelization is somewhat contradicted by the comic book Age of Rebellion: Luke Skywalker. Set between the events of The Empire Strikes Back and Return of the Jedi, it features a black-clad, green lightsaber-wielding Luke Skywalker on a mission to raid an Imperial refinery in the Outer Rim. While it's still likely that Luke went and built his lightsaber on Tatooine, this changes the timeline somewhat, as adding a whole other mission before the rescue of Han Solo means there's a much bigger gap (at least months) between completing the green lightsaber and its first on-screen appearance in Return of the Jedi.
