The best explanation for the observation that the mother found brown solids in the bottom of the mug after the cocoa had cooled 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 more easily in hot water due to its higher solubility at elevated temperatures. However, as the cocoa cools, the solubility of cocoa in water decreases, leading to some of the dissolved cocoa becoming less soluble and precipitating out as solid particles. This explains the brown solids at the bottom of the mug.