Since this started with something I said....
I find it interesting that most swords from literature pre-dated their owners, most are mythical and were given names by someone generally unknown as they are often known through legend. While LOTR follows this to a point, Sting was an exception (I believe). In other worlds (Song of Ice And Fire, Swords follow the mythical name with at least one noted exception, Joffrey names at least two of his swords and, well, he was unworthy of both.
So here it is, as you are building your persona, naming a sword also requires a sketched persona, most swords were given a name based on their deeds and ego. Swords in literature were just as good or evil as the characters who used them. Some names would have great fantasy value but bad timeline value (soul stealer may be looked at as a demonic blade by the church and its bearer a heritic).
The Samurai held there swords in religious honor, under their belief the sword had a soul in its own right, and therefore would be named. Most were given names originating in elemental, earth, or nature. As Samurai lived to serve, the sword was an extension of that service with blades handed down for many generations.
So while I make no comment on what to name your blade, its part of your chracter, I do find it an interesting topic, and a great way to add interest to your character, and a natural thing to do for something you depend your life on.