I believe I can answer your question. I will make some assumptions & you can guide me if I am wrong.
Let me start with suggesting you read a topic that is along the same line as yours. It is different but may further answer some questions on how to deal with contacts.
Post Title =
Contact list syncing with multiple people with the same name.
I am assuming your contacts are through Google & are syncing. You can have multiple individual contacts with the same home number. That is not your actual problem, correct? Your problem is = When someone calls you from that number, you want it to show up as the family name instead of an individuals?
I don't know what phone you are using or contact app, but for most the Caller ID pulls up the 1st name with that number from your contacts list. The simplest thing I have found is to create a contact "A-Smith Family", "1-Smith" or "All Smith". Some tweaking may be required to find your preference. It's easiest to start with just a family or 2 as testers. The goal is to have the contact with just the home # & Family name to appear 1st in the contact list. Then when someone calls from that Home # the "Family Contact" should show on ID. This is how I have some business contacts from the same company set-up. I get the generic company name if one calls from the company but that same number is listed for each contact too. I use "1- ____" mostly.
Does this make sense? I hope I explained it okay. It really is pretty easy. Just copy & paste the #. Someone else may have a better idea or a non-native app, but I try to stay away from duplicate apps for native programs if I can. Let me know if I misunderstood or was unclear. Please let me know if it worked or if you found something better that was not posted.