What about someone who has two lightsabers? For convenience's sake, I could easily see someone giving them names, if only something simple like "lefty" and "righty" or "lefty" and "shorty." Or a name based on overall color, such as "steel" or "black handle." Even though they might not seem like names, and might not be names as far as the owner is concerned (attachment... ), to the uninitiated masses they can become names.
Objects that have characteristics can easily become named after those characteristics. The Porsche that I had in my youth had a tendency to backfire, hence it became Chitty Chitty Bang Bang. My wife and I each owned Ford Escapes at one point. Mine was yellow. Hers was red. When the kids asked which we were going to take someplace, at first it was "yellow" or "red", and eventually they became "Mustard" and "Ketchup."
And what of someone like General Grievous? Did he call his lightsabers after the dead Jedi who once owned it? While I haven't given my Force FX lightsaber a name, I do think of it as "the Fisto."