Well, 'the family clock fanfiction' could be centered around a family heirloom which is a clock. Maybe it tells the story of how the clock has been passed down through generations, and along with it, certain family secrets or traditions. It could also involve the characters in the fanfiction having adventures related to the clock, like trying to fix it when it breaks and in the process uncovering something unexpected about their family history.
In the 'family clock fanfiction', it may be that the clock serves as a symbol of time for the family. It could be a device that somehow controls the flow of time within the family's world. For example, when the clock stops, strange things start to happen to the family members. It could also be a story about how the family tries to keep the clock running in order to maintain the normal order of their lives, and the various challenges they face in doing so.
I'm not sure specifically as there could be many different 'family clock' fanfictions. It might be about a clock that has some special significance within a family, perhaps it's a magical clock that affects the family's destiny or it could be a clock that holds memories of the family's past.