The correct option to complete the sentence is "on": We have a family mutiny on our hands.
The word "mutiny" means a rebellion against authority or a refusal to obey orders. When we say "on our hands," it means we are dealing with something that is causing us a problem or difficulty.
Therefore, "on" is the correct preposition to use in this context because it means the mutiny is happening within the family, and we are responsible for dealing with it. The other prepositions ("for," "from," and "of") do not convey the same meaning and would not make sense in this context.