I picked not visible from earth, but was more thinking “It can mean whatever you want it to mean.” 
I picked “the part of the moon where it is night”. The first thing I thought of was the Pink Floyd album, but of course that’s not all I know about it.
The evil side of the moon.
The second half of a “trip” when you are coming down and feel far worse than before you dosed yourself with chemicals.
I understand it to mean “the side of the moon not visible from Earth” even as I understand that it’s a misnomer because that side is lit by the Sun exactly as often as the side we can see.
re: the Pink Floyd choice…let us not forget the Jerry Driscoll’s closing words on the album–
DTC certainly remembered them ;).
First thing that pops into my head is “Which one?”. Context is everything.
The OP question is ambiguous. I started to scroll down to see if there were any further clarification, when I saw "Please vote before reading posts or checking wikipedia."
Ok, my interpretation of the OP: When you use it, what do you use it to mean?
In that case, unless specifically referring to the Pink Floyd album, I always mean “the side that isn’t lit”.
I could come up with other interpretations of the OP question that would have required a different answer, but that’s the interpretation I chose when forced.
I think it’s the half of the moon not lit by the sun, but the answer choices don’t match that.
The (formerly) unknown part of the Moon, i.e., the side facing away from us. In the same sense that (formerly) unexplored areas of Africa (to Europeans, anyway) were known as “darkest Africa.” Nobody thought that it was actually perpetually night in darkest Africa, except in cartoons.
It means that it’s time to queue up The Wizard of Oz again.
The Dark Side of the Moon is the side that eventually pushed it into the ways of the Sith.
(but I voted “night side”. I thing of the part we can’t see from earth as the “far side”.)
How does “The part of the moon where it is night” not match that? 
This phrase has always bothered me, because its meaning is, in effect, ambiguous. But I’ve personally interpreted it literally as the side of the moon that’s dark – i.e., in shadow – whether it’s visible from the earth or not.
There’s an option you didn’t list… “Dark Side of the Moon” is an excellent 2005 trance tune from Ernesto and Bastian (the Dogzilla Pure Filth remix is the one to go for).
Thanks for the replies so far. I always thought the phrase meant the far side of the moon. I looked through a bunch of dictionaries, and reference books on phrases, but couldn’t find mention of it. I think the phrase originated when dark meant unknown and most likely referred to the fact that we could never know what the far side of the moon looked like. Now we do know what the far side looks like, so the phrase is obsolete.
The replacement and more correct phrase “far side of the moon” may cause problems in the future. As a spaceship approaches the Earth/Moon system which side is the “far side”. The side closest to the ship or the side furthest from the Earth? Maybe we should call the hemispheres Earthside and Spaceside.
And as one of my smart ass friends pointed out, a globe doesn’t have sides.
I did read his post, but I felt I the need to give Mr. Driscoll credit 
I use it to mean the side of the moon that never faces the Earth; even though it receives sunlight it’s always “dark” to us.
Random thought: when the earth is between the moon and the sun, isn’t the entire moon dark?