I sort of agree with GG here but my personal experience may be clouding my judgement. I certainly wish that we had been a lot more informed about the consequences that can arise from not enforcing house rules but thats another story.
I'd remove the PC - if it is his only thing as you say then it will have an immediate affect on him and he may - I use that term lightly based on personal experience - realise that its a rule that has to be obeyed and there are no deviations.
Sadly the consequences of his actions so far - assuming he is using spoofing and even darker techniques etc - could be far worse down the line than a few days of conflict at this stage.
This is of course my personal opinion and not all teens are the same.
edit: Just re read the thread and see you could try taking the powerline (the ethenet connector) box from him but it only works if you are in the house as he can just go and get the other one and use it.
The only possible technological solution to is to invest in a proper replacement router.
For example the device I have can be set to restrict devices on an individual basis by time. Each device can have different times set for each day.
When the OFF time arrives the connection is killed whether wired or wireless. The device is then unable to reconnect until the start time is reached. You can restrict new wireless devices connecting to the router simply by disallowing new devices in the settings, they then have to be added manually.
There is no practical way of restricting access to web sites. Simply by downloading and using TorBrowser you can bypass router blacklists and ISP network level restrictions. So unless you are in the same room and can look at what is on the screen you will have no way of preventing inappropriate content.
As Keith has pointed out, spoofing or changing IP and Mac addresses will get round the access controls. Even if you block new connections all he needs to do is spoof the Mac address of a device that is allowed access. There are many sites on the Internet giving instructions how to do that which I'm sure any computer literate child will be well aware of.
Billion and FritzBox I know for sure work like this, though they are not the cheapest, other makes no doubt have similar options. You can download user manuals for routers and look through to see if they have the settings you need.
You may still need to monitor the connected devices in the router now and again to check for unauthorised MAC addresses and either block them completely or add to the restricted list.
Restricting tech savvy kids can be almost a full time occupation. If you decide to get a new modem/router get one that supports VDSL.
It does not matter if it has a static IP address or not and it is actually irrelevant as regards access controls on the BT hubs which I believe use the Mac of the device and not the IP address.
All the access controls are relatively easily circumvented regardless of which router you have. You are trying to herd cats.
Only problem would be that the static IP can be changed again.
You would need to set some heavy duty user restrictions on his PC user account. Something that would only allow system changes and software installation by the Administrator. Rather complicated and time consuming for most people and a bit of a learning curve.
The issue will most likely be complicated further by IPv6 IP addresses, that is something I would need to research. IPv6 can't be static when your ISP assigns dynamic prefixes as BT and the rest (to the best of my knowledge) of UK ISPs.
Some additional thoughts.
The technical one, set up a proxy server which shares it's Internet connection to that device. Basically acts as the middle man to say what and when data can be communicated over the Internet. Would need to keep an eye on the Ethernet cables.
This is the sort of solution used in Educational establishments.
Or a simpler option - install some parental control software on their devices which will allow you remotely manage when they can use the Internet and what they can go on.
Also bear in mind with the mobile devices that they can also receive and send data over the Internet over the mobile network, so limiting your own home Internet connection is one thing but doesn't cover devices that can also communicate over the mobile network.