This is related to the current discussion in my TFA edit, but if I ever get around to making this TLJ edit, one of the changes would be to make Ben’s first lightsaber the Anakin/Luke saber.

This would solve the mystery of how Maz got the saber even without the visual of her getting it from Leia, and it is a story that could be entirely inferred from in-movie cues:

The saber could have been found by Lando or Luke and given to Ben, but since we know Ben is obsessed with Vader’s legacy and collecting artifacts of that era, he would probably have sought out the saber on his own during his training.
TFA establishes that the saber can impart troubling visions of the past, so if it was in Ben’s possession during his training it could have been used as a conduit by Snoke to turn him to the Dark Side.
TLJ implies that Ben was so short on time that he didn’t search the rubble of his hut to make sure Luke was dead, and this could explain why he leaves the saber behind if it was also lost. The vision and flashbacks show Luke with his hand on R2, which may imply that Luke found the saber in the rubble and placed it inside R2 like he did in ROTJ.
R2 being with Leia confirms that Luke at least sent the droid back before going into exile, and if the saber was inside R2 it may have served as a sort of apology, one that Leia couldn’t face.
Since the saber can ‘call out’ to individuals, there is reason beyond the emotional one for Leia to want the saber far from herself, although she probably wouldn’t want to discard it entirely, hence why she would give it to a trusted and stable friend with knowledge of the Force such as Maz.

And this is where the movies pick up the story, though there are some reasons beyond this that I like the change:

Kylo Ren’s saber appears unfinished and unstable, implying that this is his first attempt at creating such a weapon. Giving him Anakin’s saber firmly establishes that he hadn’t yet constructed his own, as well as emphasizing his youth during his training and confirming that his Jedi skills were not yet complete.
Kylo specifically says that the saber belongs to him on Starkiller Base, which would be literally true with this change. Its ‘resistance’ to his power in that scene could be a more marked metaphor for how its ownership has changed on a deep level.
Luke’s heartbroken look upon seeing the saber, and his rejection of the blade, is made more powerful if it is the literal representation of his hope in and ultimate failure of his nephew. His using the image of that blade in his final confrontation with Kylo would more strongly illustrate his acceptance of the past and his greater legacy.
The image of the broken saber at the end of TLJ again more strongly resonates as the symbol of Luke’s broken Jedi Order and the enormity of the work left to do.