In A New Hope, when Darth Vader and Obi Wan engaged in a fight, Obi Wan was killed. At that time, his lightsaber fell there and Luke didn't retrieve it.
So, what happened to it? Was it destroyed?
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In the movie canon we don't really know the fate of Obi Wan's lightsaber yet. And this is even his third lightsaber.
But remember it's not even canon now and Anakin Solo might not even exist in new continuity.
In April 2014, Lucasfilm announced that all previously released expanded universe content would be declared non-canon and rebranded as Star Wars Legends. - Wikipedia
So the old canon answers don't apply anymore and new canon didn't give the answer yet.
In movie canon, the last time we see it is after the dramatic conclusion of the duel between Darth Vader and Obi-Wan Kenobi. As Darth Vader strikes his final blow, Obi-Wan turns into a Force ghost, leaving his clothes and weapon behind. We see the lightsaber last among his robes, as an unbelieving Darth Vader examines it by kicking it with his foot.
We don't see it afterwards, unlike Luke's first / Anakin's old lightsaber that seemed lost on Bespin but turns up at Maz Kanata's.
We may speculate that Darth Vader picked it up, but there's no evidence that it survived the destruction of the Death Star.
I am pretty sure that Vader took it and left it inside some safe or in his quarters.I do not believe he had it with him when he boarded his tie fighter, prior to the Death Star being blown to bits.
If the light-saber was not destroyed by the explosion and is still out there, it is floating in space somewhere in the outer rim by now.