As you have discovered, when eBay restricts an account they will also restrict any other account holder that they associate with the restricted account.
This is to prevent restricted members from simply starting up again by using a family member's name or account.
Your family members can appeal the decision, but it's unlikely to succeed. The innocent often suffer from the actions of others. There is nothing unlawful about it, in the user agreement you all accepted eBay's right to restrict any member's access to its services.
