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 remove names. Many languages don’t have separate names and work just fine (Japanese, for instance)
* technically Japan has month names, but no one uses them at all in daily life. My wife can’t even list them all right now, heh.