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.
I would second the motion to a 13 month calendar of all 28-day months (despite the fact you KNOW my birthday is going to end up on a forever Monday). The days of the week are named after scientists of seven different disciplines, the months after artists. There is a theme with the artists, so that 1/4 is musicians, 1/4 visual artists, 1/4 writers, 1/4 builders.
Whenever I see a street or city named after a politician or military officer, I always wish there were more things named after people that are actually useful for the world.
For reference, I was born on the 12th day of the month of Lady Gaga, of course an Einsteinday.