There are many mysteries in the Star Wars universe that have either never been solved or were addressed in Legends and thus ended up being considered non-canon, as is the case of the fate of Obi-Wan Kenobi’s lightsaber after his duel with Darth Vader in Star Wars: A New Hope. The history of the Star Wars universe began in 1977 and has been in constant expansion ever since, covering not only film but also television, comic books, novels, video games, and more. Although the material outside the movies is complementary, there are some that are now considered non-canon, creating confusion among fans and leaving various mysteries and plot holes.
Following the acquisition of Lucasfilm by Disney, all the Star Wars media outside the Skywalker Saga produced before April 2014 (with the exception of The Clone Wars) was deemed non-canon and labeled Star Wars Extended Universe, which Lucasfilm later rebranded as Star Wars Legends. This extended media had added to the events of the Skywalker Saga for years and even answered some of the biggest questions left by the movies, but this change in the Star Wars canon ended up affecting the information fans already had, and many mysteries were left unsolved. Such is the case of the fate of Obi-Wan’s lightsaber after his duel with Darth Vader, which was answered in Legends but the new Star Wars canon has failed to.
Star Wars: A New Hope was the audience’s introduction to this universe and the characters that would lead the sequels, the prequels, and the sequel trilogy, such as Luke Skywalker, Leia Organa, Han Solo, Darth Vader, and Sheev Palpatine. A New Hope also introduced Obi-Wan Kenobi (who had been living under the identity of Ben Kenobi), though he didn’t stay for long in the original trilogy. Obi-Wan engaged in a duel with his former apprentice, Darth Vader, who killed him on the spot, and the story continued its course after that. This left the big question of what happened to Obi-Wan's lightsaber after the duel, which in canon is simply assumed to have been destroyed along with the Death Star, but in Legends its fate was different.
Following the duel on the Death Star and Obi-Wan becoming one with the Force, Vader took the lightsaber and stored it as a trophy. Sometime later, after the Jedi Order was recreated under Luke Skywalker, some of his students, led by Tionne Solusar and Ikrit, arrived at Bast Castle searching for relics. There they came across Orloc, a false Darksider, who stole the lightsaber and threatened Ikrit with it. However, Orloc was unfamiliar with the passages and traps of Bast Castle, which nearly got him killed, allowing the Jedi to reclaim Obi-Wan’s lightsaber. It was then taken to Luke’s quarters for examination but was stolen again now by Uldir Lochett, a non-Force sensitive human who tried to become a Jedi. Uldir met Orloc in hopes that he would teach him how to use the Force, but Orloc was confronted by Uldir’s fellow students. Tionne Solusar retrieved the lightsaber, which changed hands several times during the battle, ending up on Anakin Solo’s hands, who then used it to destroy Orloc’s costume, which allowed him to mimic Force abilities, and exposed him as the fraud he was.
Obi-Wan’s lightsaber was then returned to the Jedi Academy, and no more has been revealed about it. Although in Star Wars canon the fate of Obi-Wan’s lightsaber hasn’t been addressed, it’s known that Luke used its design as the basis to build his second lightsaber, and that’s the last thing it’s known about it. With the Star Wars universe in constant expansion, it wouldn’t be surprising if new canon material finally solves this question, though it's also likely that it will never be, more so now that the Skywalker Saga has come to an end.