An alternative to the many great ideas here:
Divide your galaxy up into 100 domains named after the highest mass object within each domain. The highest mass object is poetically treated as the cosmic "ruler" of that domain, much like we anthropomorphise mountains and such on Earth. These domains can vary in size based on the gravitational influence of it's cosmic ruler. You can even divide them further, much like a feudal system of Kings, Dukes, Counts etc.
From there just use vector 3 in light years relative to the cosmic ruler. e.g.
Traveller: "I come from Uddakar alt-Selim in Archaeon."
Officer: "I'm sure you do, coordinates?"
Traveller: "103-209-77, if you really must look it up"
^ So Uddakar is the traveller's planet, "alt-" means "of", "Selim" is the local name of the star and "Archaeon" is the galactic name of a black hole that influences 1 billion stars around it and was therefore categorised as a "cosmic ruler".
This allows you to blend some cool lore and cool names with some acceptable hard science. You can lower the number of cosmic rulers to be easier and really just treat naming of planets and stars as a local thing only, where ruler and vector 3 is the universal language.