Anakin Skywalker's blue lightsaber is easily one of the most iconic items in all of Star Wars. But before the sequel trilogy released, it was assumed to be lost forever on the planet Bespin. This led to the creation of Luke's green lightsaber in Star Wars: Return of the Jedi. However, very little is known about when and where it was constructed. And so, a convincing theory suggests Luke used the kyber crystal from Qui-Gon Jinn's lightsaber to help build it.
Return of the Jedi's deleted scenes show that it was originally going to open with Luke's lightsaber construction. As he sits in a cave on Tatooine, Luke's screws the final adjustments to his new lightsaber, and then leaves for Jabba's palace. But in the end, the scene was cut, and the reveal of his green lightsaber was left for his grand escape from the Sail Barge.
While the deleted scene shows some slight adjustments to the weapon, it's known that constructing a lightsaber is no easy task. As seen in Star Wars: The Clone Wars, young Jedi Padawan would be taken to the planet Ilum to find their kyber crystal. These crystals help power the Jedi's weapon, and determine the blade's color. And as with much of the Jedi's training, this task would be a trial for every Jedi, as they had to sense the crystal that was right for them.
However, the planet Ilum was eventually overrun by the Empire, and much of it was mined for resources. Later, in Star Wars: The Force Awakens, the planet became the home for Starkiller Base, and was destroyed by the Resistance. But even before then, it's likely Luke wouldn't have been able to secure a new green crystal from Ilum, due to the whole Empire searching for him. But that leaves the question, if Luke didn't get his kyber crystal from there, then where did it come from? And so, a theory suggests that the crystal was actually taken from the lightsaber of Qui-Gon Jinn.
In Star Wars: A New Hope, Obi-Wan grabs Anakin's lightsaber from an old box in his hut. It's unknown what else Obi-Wan had in there, as he could only bring the essentials and things that held sentimental value. Meaning it's entirely possible that he kept the lightsaber of his old fallen master. And if this were the case, Luke could've easily returned to Obi-Wan's home and extracted the green crystal from Qui-Gon's weapon. This would both explain how he managed to find a kyber crystal without being detected, and why he swapped to a green blade instead of blue.
Within the old Star Wars canon (now called Legends), this theory was backed up by the Shadows of the Empire novel, which said Luke returned to his mentor's hut on Tatooine to build his lightsaber. And while this no longer fits in the Star Wars timeline, much of the formerly canon material is plucked for the new one. Which would be fitting, as not only would this explain a piece of missing lore, it also creates another link between Luke and the prequels, as Qui-Gon was the only Jedi who supported his father joining the Jedi Order.