Personally, I would go for naming them after the twelve largest moons in the solar system, since they were originally (and still are in many cultures) based on the lunar cycles, giving the months Ganymede, Titan, Callisto, etc.
What would you rename the months?
You know what… I’ve never seen this before, but I fully endorse this solution. This is mostly what we already have in place, but favoring consistency rather than flattering Roman emperors.
For those who didn’t know,