[SalesForce] How to copy standard Opportunity objects into a custom object

If you want on a button call go for custom button with apex behind it doing job of creation and populating the data.

If you want that as soon as a new Son record and automatic family record is created and populated go for trigger,

//apex method pseudo code

public pagerefrence createfamilyrecord(){

 Id recordId = System.currentPagereference().getParameters().get('Id');

 List<Son__c> lstson=[Select Id ,s_name, s_street from Son__c where id =:recordId];

 if(lstson__c.size()>0){

   Family__c fam=new Family__c();

   fam.field1=lstson.get(0).s_name;

    fam.field2=lstson.get(0).s_street;

    insert faml;

    return null;//Return where you want the navigation to be 
    }

   }

//Visualforce pseudo code

<apex:commandButton action="{!createfamilyrecord}" value="NEw Family" id="theButton"/>

You mention that you have a lookup between the objects, assuming you have a lookup to the order object on the opportunity, you just need to reference the order object in your creation of the new opportunity

Opportunity Opp = new Opportunity(
      Name=OrderObj.Order_Name__c+'-Opportunity',
      CloseDate=OrderObj.Close_Date__c,
      StageName=OrderObj.StageName__c,
      Order__c = OrderObj.Id
);