If your goal is to monitor and/or control your son's Internet access while he's using a computer at home, then your single best point of control is your router (or modem/router combo).
The reason the router is best is because all your home network traffic must go through it, with few exceptions.
Any decent home router has white/blacklisting options, as well as site/access logging. What you'd likely be interested in is the blacklisting (blocking specific websites), and logging options.
Blacklisting allows you to set sites that can't be accessed through your router. They should at least let you block specific sites, such as
www.parenting.stackexchange.com. They may (and should) let you do it at the domain level such as blocking all stacks on this site by blocking
stackexchange.com. Ideally, they'll let you block with wildcard filters, so you can block sites with keywords, such as
stack. Arguably more useful filters would be
In most cases, I think it's fine to just be upfront with your child about expectations of his Internet usage, and giving some trust that way. However, I think you, as a parent, have an obligation to ensure your child isn't engaging in illegal or illicit activity. So, blocking his access to sites that are primarily used for pornography or pirated content are up to you to block.
I'm not interested in giving a moral evaluation of whether or not the laws are worthwhile, or whether or not your child should have access to those places. That is a decision that's up to you and your household. However, I would add that if your child does anything illicit on your Internet connection, you would be culpable. For instance, if someone uses your Internet connection (even an unauthorized user) to pirate a new movie or TV episode, you may get a piracy notice from your ISP. If you get enough of these notices, your ISP may ban you from their service.
Now, back to the logging feature. Most routers I've used have at least a basic logging feature that tracks which connected devices access which websites. However, few of them have tracked the access very far back in time. This means you'd have to periodically log into the router and check it while your son is online, to see what types of sites he's accessing.
If you decide to go with router-level controls, you'll need to do two more things:
- Set a secure administrator password for the device (and not use the default)
- Control physical access to the router, as a hard reset will wipe out your settings (and reset the password the default)
Personally, I would recommend doing these things even if you didn't have a teen in the house. It would also work for guests or other family members that need to use your Internet access.
As to the claim that your child could just use their phone, that's not necessarily true, and depends on your provider. For instance, Verizon Wireless provides free content filters, advertised as Family Safeguards & Controls, that could limit what they use their data plan for.
So, with a little bit of effort, and hopefully the right router hardware/software, you can do a lot to mitigate the "worst" your child has access to. However, any methods beyond these basic ones will require increasing effort, and likely specific research for your specific needs.