Can there be a three-moon system where only two are visible most of the time? I am trying to build a moon system for a sci-fi work in collaboration with my boyfriend. He's the main writer and wants there to be three moons circling the planet in a way that most of the time, only two moons are visible and seeing all three or none of the moons is equally rare. I don't have much training in astrophysics (only a university level intro class to it), so my main consideration was that, in the case of Earth's moon, it is only fully/almost non-visible about 10% of its rotation (2-3 days of 28). What else should I consider? If it's possible, how long would each of their orbits be?

For reference, the planet would be orbiting around a Sun-like star and be about 1.13AU away from it (still within the habitable zone, but with a slightly longer year). Each of the three moons rotates on itself as fast as it rotates around Earth, just like Moon does.
Start with some basic calculations.  At any time, you can see a hemisphere, which is (gosh! :-) ) half of all space.  This means that no matter what you do the average presence (lit or not) at any time is  1.5 moons.  Since there's only one "sun," all moons present in the observable hemisphere will have an equal amount of the observable face illuminated.   

Now, if we decide to have two of the moons orbiting each other in a binary fashion (and their net center is orbiting the planet), then 50%  of the time  these two moons will be in your visible hemisphere, and 50% of that time so will the third.  Conversely, 50% of the time these two moons will not be in your visible hemisphere, and 50% of that time neither will  the third.   

All of the above assumes orbits similar to our Moon, i.e. roughly equatorial, around our own planet's rotational axis.  What if these 3 moons somehow manage to be circumpolar?  In this case they'll be permanently "quarter moons," visible whenever in our hemisphere of view.  Now make the third moon circumpolar with such huge ellipticity that it's not visible to the naked eye for most of the orbit. 

So the only thing not possible in either  of the above scenarios is managing to make the first two moons visible for "most" but not "all" of the time.   Perhaps someone else can come up with a more esoteric solution.

Note: all this assumes the moons are small enough that their interactions do not lead to short-term chaotic behavior.