There isn't a lot of evidence to support this theory, yet. However, there is some.
1. There sabers look alike.
The sabers power button is on the same side. This indicates that the saber was just modified.
Another thing: their bases. The bases if the sabers are alike, just Kylos is darker.
2. Kylo has Vaders helmet, why not the saber.
If Kylo has Vader's helmet, why wouldn't he have his saber. As he says, "I will finish, what you started", this indicates that Vader is Kylos hero, and so he would have the saber of his hero.
That's pretty much all the proof there is for this theory so far, but more is to come. May the force be with you...