I'm gonna be a jerk here, and I'm sorry, but is it really a moon if it orbits a star?
Perhaps technically, no; things that orbit stars are "planets". However, we still don't really have a good definition of what constitutes a "star". Brown dwarfs are technically "sub-stellar objects" and not stars; however, brown dwarfs are considered "stars" from the point of view of the ED stellar forge, which lists them on the galaxy map info screen as "stars" and uses the generic star icon to depict them on the main screen. And currently, the astronomical community uses the word "planet" to describe an object discovered orbiting a brown dwarf.
However, the stellar forge does sometimes place "stars", even main-sequence stars, in "planetary orbits". Such objects are considered "planets" by the game, even though they are described and labelled as stars. For example, only stars in planetary orbits can be given rings. And any "planets" orbiting a star in a planetary orbit are labelled as if they were moons, not as if they were planets, as you can see by the picture roboteconomist posted above; the highlighted world is labelled "Hegeia VE-P d6-6 4 d", or "the fourth moon of the fourth planet of the Hegeia VE-P d6-6 system".
Since we haven't yet discovered any in-real-life things orbiting a brown dwarf that in turn orbits a star, the IAU has not had to make a decision on whether to call such objects "moons" or "planets". I would propose that using either word is acceptable in this circumstance.
