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?
I’ll second the IFC. My only addition would be that the Year Day and the Leap Day are mandatory holidays: fucking everything except critical utility infrastructure shift/on-call work (police, fire, power/water/sewage, etc) is closed. Employers must not require an employee to work consecutive Year or Leap Days.