The best explanation for the brown solids seen in the bottom of the mug is: The cocoa was mixed with hot water, but some cocoa settled as it cooled because the solubility went down.
When hot cocoa is prepared, cocoa powder dissolves in hot water. As the temperature decreases, the solubility of cocoa in water typically decreases as well, leading to some of the dissolved cocoa coming out of solution and settling at the bottom of the mug. This phenomenon is related to how temperature affects the solubility of solid substances in liquids.