I named my first saber "Perseverance" - mostly because I had some terribad luck when trying to build it. I'd explain how... but I think I'd get chased off the boards by an angry mob with torches and pitchforks. (PROTIP: When they say "start simple," it really is a good idea!) And over the nearly two years since I built it, I've had to rewire it completely twice, perform a half-dozen spot repairs, and had the LED pads on the Ultrasound it had in it short out and create the magic smoke... right at the stadium where I was about to use it for a Rebel Legion event as I was trying to show it off to Jay-Gon. (And thus proved why it's always a good idea to carry spare lightsabers!)

As far as the overall tradition... I remember in I, Jedi when Corran constructs his lightsaber, and there was mention about the rituals surrounding it. I don't have my (well-used, disintegrating) copy at hand, but I do remember how it made mention of a final step, as the power cell is charged for the first time, how the Force is used to perform a final "seasoning" of the electronics, making them slightly more than just a pile of parts and crystals. At the same time, mention was made of Corran feeling changes being made in him, truly starting to stand on his own as a Jedi and heir to the Corellian Jedi heritage. I suppose that, especially for the more combat-oriented Jedi, that sort of deep connection with a weapon could motivate one to give it a name - the name of a comrade or loved one, a virtue or ideal to uphold, or something like that.