I'd like to find a way to sync the lunar cycle and solar cycles since the earth's, moon's, and revolutions around the sun are soooo close (5 days off) plus it'd make sense to keep in theme with the Babylonian-esque base-60 system (where 60 is readily divided and a factor of 360 days, 12 months, 30 days, etc).
I'd personally prefer 12 months with 30 days each, a 6-day week (makes for even rotations in shifts, 4 on 2 off), and an inter-calary week of 5 to 6 days at the new year.
If we're going for broke on this I'd also want to convert to the dozenal system over decimal, as 12 is more easily divisible by smaller numbers which means easier division for numbers we use more often (like 3 or 4), which means that ¼ would be 0.3 and ⅓ would be 0.4.
you can still use your fingers. it's how we got our standard of time. Back then they counted the joints in our fingers minus thumb. 4 sets of 3 for our four fingers and 3 joints per finger. Then 5 sets of 12 to make 60. as they would use the fingers on the other hand to track how many times they counted to 12.
A dozenal system is more difficult in multiplication. Decimal: 10^7 =10000000, 10^8=100000000, 10^9=1000000000, etc.
Dozenal: 12^7= 35831808, 12^8=429981696, 12^9=5159780352.
Gets very messy very quick.
I would say that at the very least we could adjust February by taking a day from July and August and the extra day every four years could be added inbetween them as a "monthless" day in the middle of the summer.