Kylo Ren seems determined to emulate Darth Vader, right down to keeping a little "Vader shrine" in his private chambers (complete with the helmet that Luke cremated at the end of Return of the Jedi).
We know that Ren's lightsaber contains a Kyber crystal that is cracked and broken. Could this possibly be the same one that was in Vader's lightsaber?
During his duel with Luke, Vader's lightsaber fell into the chasm in Palpatine's Throne Room, at the bottom of which lies the Death Star II's 'reactor core'.
Per the (fully canon) Ultimate Star Wars factbook:
In their final duel, Luke momentarily succumbs to his anger and cuts off his father's hand. Thus Vader's lightsaber falls into the same energy well in which Sidious perishes.
Per the Official Novelisation:
The hand, along with bits of metal, wires, and electronic devices, clattered uselessly away while Vader’s lightsaber tumbled over the edge of the span, into the endless shaft below, without a trace.
Per the new Junior Novelisation:
But Luke slashes with his saber, slicing Vader’s arm off. The metal limb tumbles down into the shaft, taking the lightsaber with it.
Note that the Death Star II exploded shortly afterwards with very few survivors. All accounts (both within the main canon and EU) have the saber being destroyed when it fell into the reactor core.
