the fixed names [...] give a sense of geography to the galaxy, allowing us to say a name and have that actually mean one, and only one, place in the universe.
Except they don't. I have on more than one occasion seen several stars with identical names in the same galaxy, sometimes even very close to each other. The name generator itself is a nice gadget that could benefit from a linguistic overhaul, but it's certainly not a pressing issue.
I hope we will get a means to name systems and stars independently from each other just like we can name planets and planet plates.
I'd love to see some automation, too: If you find a star you like, scan the system, and claim it, you get to rename the star if you so desire, and the system itself could inherit the name of the first star, unless or until you rename either the system, or the star. The planets would automatically be renamed <first star's name> I, II, III, IV, ... until you claim each, and again can give them individual names.
Binary or multiple star systems should allow for each star to have its own name, or share <name> A, B, C, ..., and optionally have a system name that is different from the star's name.
System names could act as aliases for coordinates, so you could set your waypoint to either "2, 2, 2" or "<insert favourite star name>"... well except you'd end up inside the star. Maybe not such a brilliant (haha) idea.
As it is currently, you can name a planet, and then continue to rename each plate individually, but the planet's name sticks, AFAICT there's no way you can't change it after the fact. Naming the individual plates is a feature I very much like and hope will be retained, because it allows for "continents" or "cities", but there should be a way to rename the planet itself, for any number of reasons - captured in war, fat finger typo, named by a troll, whatever.
Likewise, there should always be a way to rename the stars, and the systems themselves. If it needs to be rate-limited, make it cost time, faction points, money, and/or resources, possibly consecutively increasing amounts.
TL;DR:
(In a claimed system,) the first star that is given a custom name grandfathers this name down the hierarchy - the system inherits the name, a possible companion star results in the first star's name and a qualifier, likewise the planets inherit the star's name and a number; everything can be renamed however many times is necessary, if necessary at increasing cost.