I liked the theory that Kylo's lightsaber was using the salvaged, cracked crystal that used to be inside the hilt of Darth Vader's own lightsaber, seeing as how he's coveting everything Vader.
He's a big Vader stan.
How is that possible seeing as the Death Star 2 blew up? Only Luke made it off that thing with Vader's body and he didn't have time to go fishing for his old man's lightsaber.
At least in cloud city we can speculate that the cleaning/maintenance droids/people/etc found Luke/Anakin's saber and kept it/sold it etc.