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.
This is the right answer. 13 months of 4 weeks and an extra day on the last month.
How does this line up with lunar cycles? Is the solar year closer to 12 or 13 months?
Edit: I decided to answer my own question. It turns out there are about 12.4 lunar cycles per solar year, so it does fall right between 12 and 13, meaning that (in my opinion) both 12 and 13 months seem like reasonable systems.
Unfortunately it also means there’s no way to really sync the months with the year perfectly.
A side note on the solar year/lunar month not lining up… I know that planets in a star system tend to fall into resonant orbits, so you’ll often see common ratios in planetary orbital periods, making occultations more common (when planets all line up). But I actually don’t know how this relates to a planet’s moons.