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?
edit: Simply using the numbers is already a thing (see, for example, 2026-01-01. January is represented as month 1), and reordering the existing month names would be a real headache, as it would be difficult to tell whether you are reading a date using the old or new naming scheme.
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.