That's a great question Constaninou!
First things first: the moon you see from the surface of the Earth is viewed through our atmosphere, so this will have an impact on the colour we observe. And atmospheric variations will result in some changes in colour.
I use daylight WB for astrophotography because it tends to give a more accurate representation of the colour I'm seeing at that time, although this may not be an accurate representation of the actual object's colour when viewed from space. AWB can just be a bit of a gamble for this kind of work.
There's also a subjective aspect here. While the brownish moon may be more accurate, I personally prefer a monochrome moon which looks grey. So I normally thow away any colour information later. Same for The Sun, which most people prefer to look yellow. When shooting the Sun, you have to use a filter which can give a false colour or a monochrome effect. Since this isn't normally that pleasing to look at, I tend to give it a yellow tint afterwards...
Which do you prefer?
As for whether the moon has any colour, check out this interesting article:
http://www.mikeoates.org/mas/projects/m ... /intro.htm