In Star Wars lore, the lightsaber is not just a weapon but a manifestation of a Jedi's connection to the Force. Thus, the weapons of the Jedi and the Sith often become reflections of their character, symbolizing their relationship to their mentors, their combat styles and even their standing within the Jedi Order or the ranks of the Sith. The original Star Wars trilogy sees the young Luke Skywalker embark on his Jedi training under the tutelage of Jedi Masters Obi-Wan Kenobi and Yoda. And initially, Luke wields the lightsaber previously owned by his father, Anakin Skywalker, before crafting his own sometime between The Empire Strikes Back and Return of the Jedi.
Unlike his father's blue-bladed weapon, Luke's second lightsaber is green in color, marking the first appearance of a green lightsaber in Star Wars canon. The real-world reason for this was entirely practical -- the weapon's original blue blade (which even made it into early trailers and onto the film posters) didn't show up well against the clear blue skies of the sail barge scene, in which Luke first uses the weapon. However, in the current Star Wars canon, no in-universe explanation has been given for the color change. Well, a popular theory suggests it may all come down to Obi-Wan Kenobi's own Jedi master, Qui-Gon Jinn.
Qui-Gon was first introduced to the Star Wars universe a full 16 years after Return of the Jedi when George Lucas launched the prequel trilogy with 1999's The Phantom Menace. The film sees a youthful Obi-Wan Kenobi in his days as a Padawan learner under this enigmatic Jedi Master who, like Luke Skywalker, wields a green lightsaber. It is Qui-Gon who, upon meeting the young Anakin Skywalker, senses the boy's destiny as the Chosen One of Jedi prophecy and becomes a father figure to him.
When Qui-Gon is killed by Darth Maul in the film's climax, however, it leaves Anakin without the compassion and guidance of a father, setting up his eventual fall to the Dark Side and his redemption at the hand of his son, Luke. And in the final confrontation between Luke and Vader, leading to Anakin's redemption, Luke wields his green lightsaber. And through that lightsaber, Luke may bear a connection to Qui-Gon himself, bringing the father/son story of Star Wars full circle.
In order to build his lightsaber, the canon novelization Return of the Jedi: Beware the Power of the Dark Side! reveals that Luke had returned to Obi-Wan's hut on Tatooine, stating, "After leaving Obi-Wan’s hut, he knew he had at last collected everything he needed." The pre-Disney Star Wars Legends continuity also saw Luke return to Obi-Wan's hut, where he forged his own synthetic crystal to power his lightsaber. However, in Disney's new Star Wars canon, almost all lightsabers are powered by natural Kyber crystals, with synthetic crystals generally depicted as unstable and explosive. As Luke found the final components for his lightsaber in the home of Obi-Wan Kenobi, it's possible he used the crystal that once belonged to Qui-Gon Jinn.
The last time Qui-Gon's lightsaber is shown in action on screen is in the hands of Obi-Wan, who slices Darth Maul in half with the weapon, avenging his fallen master. While the weapon is not seen again after this moment in any of the movies, an officially licensed Obi-Wan figure depicts the exiled Jedi Master carrying his former master's lightsaber, alongside his own and Anakin's. While the canonicity of such an action figure is not entirely clear, it is the only explanation that has been offered as to the fate of Qui-Gon's lightsaber after his death.
Building a lightsaber is a vital part of a Jedi's training, owing to the spiritual connection between the weapon and its wielder, so it is unlikely Luke would have simply used Qui-Gon's lightsaber rather than building his own. However, by the time Luke had started his Jedi training, the Empire had established an iron grip over Ilum, the planet where Jedi traditionally found their Kyber crystals. In light of this and other Imperial legislation aimed at preventing potential Jedi from obtaining lightsabers, Luke may have had to use the crystal from Qui-Gon Jinn's lightsaber if it had indeed been kept by Obi-Wan.
As well as explaining the green coloration of Luke's second lightsaber, the use of Qui-Gon's Kyber crystal would create a spiritual link to Anakin's would-be master. Luke's final confrontation with Vader, in which he saves his father from the Dark Side while carrying his green saber, would therefore see Qui-Gon's presence felt at both the beginning and the end of Anakin's Jedi path.