Apart from the familial connection, I think it makes sense for Kylo Ren to want it given that Anakin's last lightsaber was also Darth Vader's first lightsaber (and was used to commit some of the worst acts we've seen Vader do on screen -- those poor younglings...). Unless he had backups at his castle, it might also be the only surviving Vader lightsaber since the other fell into the Death Star core along with his mechanical hand.
Originally and we know can confirm this as true. There was a scene with kylo and the knights pillaging a village. And some how the graflex got into the hands of the clan leader. This is now part of the Rey flash back scene when she touches it
You can actually hear the graflex ignite, then kylo’s ignited and you see kylo stab him.
There is a scene that was filmed where the clan leader was on the ground with the graflex (in the mud and rain, I forget if kylo kicks him.. to check if he’s dead)
and before kylo bends over to pick up the graflex, Maz jumps in, steals it and goes on the run
Some how kylo found or heard of this lightsaber and tracked it down.. don’t forget during the rule of the empire owning a lightsaber was illegal (canon wise) and if found were suppose to be surrendered to the empire and they destroyed them in a public ceremony
So by this time in the Star Wars universe lightsabers are even more rare..
Anyway.. look at me babble. This explains why when Finn ignites the graflex kylo yells “that lightsaber, it belongs to me!!”
I knew when I first saw it in the theater something had to be cut.. they left something important out about the connection between kylo and the graflex.. years later we finally found out
... that rain scene has like 3 different origin stories.. the last one we see is what we got. But there are a few other parts filmed that we never got to see..