"My family has a big house." The subject is 'family', which is singular. "The members of my family have a big house" The subject is 'members' which is plural. You would almost never hear it the second way. More likely, since the individual members would probably own different houses, it would be: "The members of my family have big houses." And if they all have big houses, your family is doing quite well. ;-)

My family has a big house.