Designing a Lunar calendar from scratch:
The most significant cyclical change on the Moon is the light/dark cycle. Any surface activities need to be scheduled around this, either during the light period (eg. construction benefits from being able to see what you're doing), or during the dark period (astronomy benefits from not having to block out the Sun). The primary divisions of the calendar would be the "month" of 2,551,443 Earth seconds (a full light/dark cycle), and the "half-month" (light or dark) of 1,275,721.5 Earth seconds.
The other cyclical change is the human sleep cycle, which is hardwired to be close to 24 hours (most people can adapt to a day of 22 to 26 hours). Dividing the "month" into an even number of days permits 28, 30, or 32 days, of which a 30-day month gives the closest to a 24-hour day (85,048.1 Earth seconds).
The year isn't a significant unit of time. Unlike on Earth, there is no cycle of seasons to tie the measurement of time to. A multiple of the "month" may exist as a unit of timekeeping, but there won't be any great drive to keep it in sync with the movement of the stars (so, no leap days).
A "half-month" of 15 days strongly suggests a 5-day subdivision, but, like the year, this is a convenience measure rather than a natural cycle. For daytime surface work, "first week" and "third week" have better lighting conditions than "second week" (when the Sun is directly overhead, leading to reduced surface constrast).
Now, the Earth second is rather strongly tied into all sorts of things, and people tend to like their timekeeping to involve integer multiples. An 85,048-second day ties nicely into the light-based month (requiring the addition of 3 "leap seconds" to keep it in sync), but it doesn't have much in the way of prime factors: you can divide it into eight 10,631-second hours, but that's it.
A better value would be an 85,000-second day: with an abundance of 2s and 5s in the prime factorization (plus a pesky 17), it could be divided into 25 hours of 50 minutes of 68 seconds, or any of a number of other ways. This in turn becomes a half-month of 1,275,000 seconds and a month of 2,550,000, requiring the addition of 1443 seconds somewhere to keep the light-based month and the timekeeping month in sync.