Do days and months on the Moon have names? On Earth we have various calendars, for example,
Days: Monday, Tuesday, Wednesday, etc., etc.
Months: January, February, March
Does the Moon have names for its "daily" rotations, etc.?
It sounds like a silly question, and I am not sure if I've asked it using the correct terminology. I suppose what I'm trying to ask is; from a viewpoint of someone living on the Moon - does it have "day" names?
 A: Since no one actually lives on the Moon, there is no call to have special names.
A day and a month on the Moon are the same length, 29.5 Earth days. Each new lunar month is called a "lunation" and numbered. Lunation 1094 started on 2011 Jun 01 and Lunation 1095 starts on 2011 Jul 01.
A: Proabably not, at least not that I've ever heard of.  Since no one has ever lived there :), there has never been any sort of calendaring system needed.  Even the longest Apollo missions were only there a few days.
I'm sure if there was ever a permanent base (or bases) there, some sort of time keeping system would be devised but it would also make sense to just use the Earth based UTC since that is the cycle the inhabitants would be on.  The lunar day-night cycle would be a little too long to adapt to.
A: As 1 lunar day is roughly 27 earth days, I doubt they'd have names of days. Most likely if there was ever a permanent base or something like the ISS, they would use earth days and months. In other words Monday on the moon would still be 24 hours long and would not have roughly half day and night like the majority of earth does. 
It would make most sense to keep in sync with the primary location on earth that they communicate to. Meaning when the people on the moon are awake would be the same time as the people who they communicate officially with. 
Until a large truly permanent settlement is created which is minimally reliant on earth, they won't have their own days or anything even then they might not still. It depends whatever is easiest. 
