Saber, Saber, Burning Bright
In some ways, the entire Star Wars saga is bound up in this lightsaber. Anakin gets it sometime between Episodes I and II, Obi-Wan claims it as swag after turning Anakin into a red-eyed charcoal briquette in Episode III, he gives it to Luke in Episode IV, Luke loses it in Episode V, and somewhere between then and now, it falls into the spindly hands of Maz.
The screenwriters cleverly deflect how, exactly, at the start of the attack on Maz's place:
HAN: Where'd you get that?
MAZ: A good question for another time.
Way to Jedi hand wave that one away, guys.
In any case, the lightsaber has seen a lot of miles. And clearly, it plays a big role here since it's calling to Rey, used by Finn and Rey, and coveted by Kylo Ren. When your three main characters have the hots for a specific object, you'd bet it's going to be important.
In this case, it's a question of the saber's legacy. It was previous owned by the two biggest heroes in the saga, and arguably the two most powerful Jedi who ever lived. And it's calling to Rey, specifically, which suggests that she's going to be just as important as they were. Kylo Ren wants it, too, but it will never be his…quite literally, since it actually flies straight past his outstretched hand in the final fight.
That's a significant snub, and getting snubbed by the Force is kind of a big deal. In that sense, the lightsaber is reinforcing both Rey's and Ren's characters. Ren is the prima donna, the weaselly middle manager who suspects he isn't quite up for the job. We imagine that losing the lightsaber popularity contest to Rey isn't going to help him much on that front.
As for Rey, she's always been the outcast, the lonely one, the one who never got any attention. She doesn't think she's very special…but she is, and the lightsaber actively goes out of its way to tell her that.
No Need to Feel Blue
And the name is key. This is Anakin's lightsaber…not Vader's. Vader's is red and menacing and no doubt fed by the blood of small children. Anakin was a much different person, and his lightsaber reflects that. It's a symbol of the better man Anakin used to be and, while yes, he did actually butcher a whole school full of kids with it, he also lost it a short time later, suggesting that the lightsaber (or at least the Force) was not down with being used like that.
We're pretty sure Kylo Ren doesn't understand that. He wants the saber because he thinks it's another connection to Darth Vader and will help purge him of those pesky ethics the Jedi keep going on about. Frankly, he's not paying attention.
Vader eventually returned to the light, after all, and while Ren keeps looking to him to provide "clarity" toward the dark side, he misses the fact that even Vader lost that clarity in the end.
That's why the lightsaber finds its way to Rey instead of him, and why Rey eventually returns it to Luke. Luke lost it in part because he needed to be his own man, but Star Wars knows better than anyone that you can't outrun the past. You can atone for any mistake, as long as you remember who you are.
Both Rey and Luke seem to need reminding of that…and Anakin's lightsaber is the perfect way to do it.