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.
Each month would be named after one of Elon Musk’s children, but the names would be randomized every year (any future children are added to the rotation). Eventually this becomes impossible to keep up with and we just end up using numbers anyway, which is fine with me.