In my opinion, I think that the question hinges on どちらか, or as the song has it, どっちか. If you looked at the link, webilo has a pretty good definition, if you can read the Japanese:
It basically says that this is an expression behaves as a pronoun. This pronoun does not specify which option is taken.
In other words, it means
どっちかの can simply be translated as the possessive どっちか (either).
Long story short, I wouldn't say that this is a specific grammar structure, but rather a combination of several grammar structures. As such, it's not uncommon, but not common enough to be listed as its own grammar stucture.
I'm going to break it down from here.
どっちかの夜: Evening in either place
My translation is as follows.
Evening/Nighttime in either place is daytime (for the other).
Note that I took liberty of assuming we were talking about different places from the clues you gave in your question. Also note that my addition in parenthesis isn't specifically stated. I suspect that there is more to the sentence than you included.