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?
They would turn into 13 months and each one would have 28 days. September to December would be placed 7th to 10th, then July and August at the end, followed by Hermes as the 13th.