So The Force Awakens has given a bit of an Excalibur/Master Sword quality to the Lightsaber that Luke inherited from his father. So let's analyze the origin of this weapon.
In The Hobbit: The Unexpected Journey, Balin tells Bilbo that swords are named for things they've done. This sets up Sting earning it's name in the next movie, but it also shows that a legendary sword can have a totally mundane origin.
That is why since The Force Awakens I kind of like even more the fact that chronologically we don't see this Lightsaber till the third movie. It's not the only or even the first Lightsaber that Anakin used. In Attack of The Clones he goes through two, and if I remember correctly both were Green. Going off the six Lucas films alone I kind of like the symmetry of Anakin starting with a Green one and then going to Blue, and then in the OT Luke goes in the opposite direction. But I'll bet there were a lot of Prequels haters who just whined "did Lucas forget that Luke's Lightsaber was originally Anakin's?"
This is the only Lightsaber he uses in Episode III (besides briefly doing s scissor move with Dooku's), the script originally included an origin for the Red Lightsaber Vader used in the OT, but the film wound up flowing better without it.
I know one Podcast review of of The Force Awakens I listened to observed how this Lightsaber had been used by the both the Darkside and the Light. Some have said you could hear the Younlings Anakin slew during Rey's vision. I think everything in that vision comes from one of two origins, Rey's memories, and the Lightsaber's.
I think it's possible that the Lightsaber choosing Rey wasn't really because she was the good guy which is the audience's natural first assumption. But it may be she's a more worthy successor to the legacy of Darth Vader as much as Anakin Skywalker.
So it started out no different then any other Lightsaber, but it gained an interesting history.