I tend to use the generated name (at least partially) for most systems I discover. However I will usually drop part of it, if it feels too clunky to me. Planets are mostly named after the star they orbit, and add one of the numbering/lettering systems below to it, then use that same naming scheme for the rest of the system.
I also know that as of this writing, there is a maximum of 6 visitable celestial bodies per system. I am going to provide more in my naming system in the hopes that eventually HG will put more stuff in each system.