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?
I would just redo the entire thing based on the same idea of 13 28 day months with extra days as holidays in the theme of the International Fixed Calendar but rename all of the months and have the year start on one of the equinox instead of a week after a solstice like the Gregorian calendar.
No, I don’t have suggestions for the names of the months, but naming them based on the solar cycle would be fun. Not sure how to make that work consistently for the northern and southern hemispheres, but if it was based on something like 'northern solstice/southern solstice and first equinox, second equinox for the months with those date and maybe waxing and waning could work.