This is most likely the carrier intentionally deactivating the old SIM.

When iPhones first came out, your current phone would normally get deactivated at the moment you purchased a new phone and told the carriers to move the service to the new phone. Now, they wait until the new phone checks in to perform the swap to account for delivery delays, backups, etc...

Your first step should be to contact the carrier and explain that you want service on both phones and arrange for which number rings to which device. They might be able to accommodate this over the phone, but if the old SIM has been deactivated in the networks, most carriers are very unwilling (or unable) to re-activate the old SIM and instead ask you to get a new SIM from them at a convenient local retail spot or by them shipping you new SIM card(s).

Apple's restore process and activation of the device doesn't affect network activation and for locked phones, you just need a SIM for the correct carrier to activate the device. iOS does not check or care that the contract is expired, not established or live for iOS activation - just that the SIM is manufactured for the correct carrier.

Set up new AppleIDs for your wife and son. This has several advantages: If they use a different computer than you, they can manage different iTunes playlists, different sets of address book contacts, etc without affecting your personal settings or preferences. They will each also be able to sync their own photos and documents and set up their own mail account through iCloud.

In Settings, sign in to your AppleID on each device for iTunes Store and to their personal AppleIDs for iCloud. This will mean that app and music purchases you make on the account will still be available on all devices, but that your son and wife will each have their own iMessage account. They will no longer receive your texts, but they will be able to send and receive iMessage texts for free over WiFi.

You can also share photos this way; photo streams can be shared between AppleIDs. If they are using your ID, any photos they take will end up on your device too. With their own IDs, this will not happen unless you specifically set it up to do so.

If you all sync to different computers iTunes Home Sharing can be used to keep purchased content in sync. Just authorize each computer using your AppleID.

One caveat: Make sure that neither your wife or son use their own AppleID for purchases, or only they will be able to see the app. There is a way to set up an AppleID without entering a credit card number; especially for a younger son, this is a really good idea.