In the question IAU 2018 Exoplanet definition the IAU's definition of an exoplanet is given which includes objects around brown dwarfs.
So the answer to your question is that a planetary-mass object orbiting a brown dwarf is called a planet.
That being said, all the exoplanet catalogs include objects with masses above 13 Jupiter masses. In other words low-mass brown dwarfs are counted as planets by the exoplanet catalogs if they orbit a star. So if they are counted as planets then they would presumably be orbited by moons. So the satellites of brown dwarfs that orbit stars could be called either moons or planets depending on whether you go by the exoplanet catalogs or by the IAU definition.