Final answer:
The phrase 'in the gym' in the provided sentence is an adverb phrase modifying the verb 'stayed', indicating where the student council stayed to clean up.
Step-by-step explanation:
In the sentence 'After the party, the student council stayed in the gym to clean up and return the decorations to the storage closet,' the phrase 'in the gym' is a prepositional phrase. This phrase is functioning as an adverb because it modifies the verb 'stayed', indicating the place where the student council remained to perform the actions of cleaning up and returning the decorations. Therefore, the correct option is that 'in the gym' is an adverb phrase modifying 'stayed'.