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?
This wouldn’t be a good idea, as then people would easily mix up dates using the old names with your system (e.g. January would go from being the first month to being the seventh!)