Because Ren is obsessed with Vader, finishing what Vader started, and quite frankly sort of has a legitimate claim of inheritance.
Abrams explained this in an interview with IGN.
“Kylo Ren idolizes Darth Vader, not Anakin Skywalker,” Abrams explained. “He idolizes what Vader represents and what Vader was trying to do. The idea that Vader didn’t succeed, if you look at it from Ren’s point of view, he was seduced by the enemy and failed because of that seduction. So the idea is that Ren wants to complete the thing that Vader started.”
Ren discusses with Snoke the fact that he, unlike Vader, is immune to the light, and will finish what Vader started. We see that Ren already has the remnants of Vader's armor, which he would probably use himself, if it were wearable.
“Kylo Ren, I watched the Galactic Empire rise, and then fall. The gullible prattle on about the triumph of truth and justice, of individualism and free will. As if such things were solid and real instead of simple subjective judgments. The historians have it all wrong. It was neither poor strategy nor arrogance that brought down the Empire. You know too well what did.”
Ren nodded once. “Sentiment.”
“Yes. Such a simple thing. Such a foolish error of judgment. A momentary lapse in an otherwise exemplary life. Had Lord Vader not succumbed to emotion at the crucial moment— had the father killed the son— the Empire would have prevailed. And there would be no threat of Skywalker’s return today.”
“I am immune to the light,” Ren assured him confidently. “By the grace of your training, I will not be seduced.”
“Your self-belief is commendable, Kylo Ren, but do not let it blind you. No one knows the limits of his own power until it has been tested to the utmost, as yours has not been. That day may yet come. There has been an awakening in the Force. Have you felt it?”
Ren nodded. “Yes.”
“The elements align, Kylo Ren. You alone are caught in the winds of the storm. Your bond is not just to Vader, but to Skywalker himself. Leia…”
“There is no need for concern.” Despite the Supreme Leader’s cautioning, Ren’s assurance remained unbounded. “Together we will destroy the Resistance— and the last Jedi.”
Between his obsession and
bloodline, which gives him a sort of rightful, if twisted claim of inheritance.
it makes perfect sense that Ren would want Vader's lightsaber.