As a parent, I was often presented with similar arguments from my kids. I can see at least two fallacies in the scenario, depending on interpretations: equivocation and denying the antecedent.
The argument is based on the ambiguous construal of the content of choice.
For the parents, the content of choice is this: "Clean your room or else (some sort of punishment would ensue)!" The child however construes the choice as follows: "I clean my room ( = I get $50: To her, cleaning implies a monetary compensation as she believes that all unvoluntary labors must be compensated) or I do not clean my room (= I exercise my free will)."
Since the child understands the content of her choice in this way, she rightfully complains that she is in dilemma: "Either I clean my room (in this case, her cleaning action lacks free will since her work without the remuneration is forced) or I do not clean my room (in this case, her exercising her free will will be construed as disobedience).
If the parents explain what is the content of the choice for the child, she would realize that she does have a choice after all.
Denying the antecedent
Upon hearing "Clean your room," the child permissibly translates the sentence into a disjunctive form. "Either I don't get $50 or I clean the room." (disjunction introduction). The disjunctive form is equivalent to the following conditional: "If I get $50, then I clean the room." The child did not get $50. Thus the child concludes that she does not clean the room. The child's reasoning however commits the fallacy of denying the antecedent.