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?
13 months of 28 days each. New Years is its own day outside of a month. Monday is first of each month. New Year’s is not a day of the week. Leap year feb 29th is moved to May to be a nice weather day for most of the world. It too is a holiday, and not a day of the week.
Name this holiday, and bonus, the 13 months.
Keep the month names the same but name the 13th “Smarch”.