You can normally add a switch into an existing network in order to extend it but that's not quite what you want to do. If I've got you right you want to leave your son's PC connected to the internet via your wireless network and then connect the Livebox to his PC via the BT Home Hub. I don't think it's possible because the Home Hub is a combined modem/router rather than just a switch.
Going back to the original problem, I wouldn't have thought the age of your router should matter. 99 times out of a 100 it's a firewall that causes the problem when connecting to a network. Have you allowed access in both directions to the Livebox?
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Do you know that we're all in line for succession to the throne? Really?
Well, if forty-eight million, two hundred thousand, seven hundred and one people died I'd be Queen.