Is it appropriate for a student to have an advisor who is also a family member? We know that family can have a lot of impact on academic direction of a student, but is it appropriate for a student to have a supervisor/advisor who is also a family member? For example, a dad might supervise their son. What would make this more or less appropriate? What conditions might be required to make this appropriate?
I'm sure that such cases might happen (and frequently do for earlier education, like elementary school), but they should not happen at the graduate student level.

All of the usual concerns of nepotism and coercion apply, just as for any other professional relationship.  Moreover, at the graduate level, there are so many different options available and an emphasis placed on intellectual "cross-fertilization" that means most institutions strongly prefer to send their students elsewhere for future positions.  I cannot then see how a graduate student supervised by their own parent would be anything but a rather dubious relationship at multiple levels.