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.
01001010 01100001 01101110 01110101 01100001 01110010 01111001
01000110 01100101 01100010 01110010 01110101 01100001 01110010 01111001
01001101 01100001 01110010 01100011 01101000
01000001 01110000 01110010 01101001 01101100
01001101 01100001 01111001
01001010 01110101 01101110 01100101
01001010 01110101 01101100 01111001
01000001 01110101 01100111 01110101 01110011 01110100
01010011 01100101 01110000 01110100 01100101 01101101 01100010 01100101 01110010
01001111 01100011 01110100 01101111 01100010 01100101 01110010
01001110 01101111 01110110 01100101 01101101 01100010 01100101 01110010
01000100 01100101 01100011 01100101 01101101 01100010 01100101 01110010