How do I get un-banned from a site?

The very first thing I would do - particularly since you're the parent in this scenario - is contact the site owner or support staff to request the un-banning. In all honesty, this is the only truly ethical approach to the situation that could result in regaining access.

Explain the situation as best you can. Be polite (anger at any perceived injustice is only likely to make matters worse), indicate that your son made an honest mistake and that it won't happen again. See if they have any flexibility in their response.

Again, they're not required to let your son back in. In fact, in an environment where people are hyper-sensitive to online predation it's very possible that they're required to take these actions if only for purposes of liability. It's possible that their lawyers won't let them reverse a decision once made.

However, if you can actually reach someone and explain an honest error it's possible that they might.

The only other truly ethical approach is to give up. Walk away having learned a hard lesson.

Bypassing bans can be very difficult, but occasionally possible. It all depends on the exact technology that the service uses to identify you.

IP address blocks are the most common. Getting a new IP address isn't always as easy as we might think. You indicated that you renewed your IP address, but in fact that's just as likely to have given you the same IP address as before. If you have a static IP, contact your ISP. If you have a dynamic IP you might try leaving whatever device is connected directly to the internet off (typically a router) for a couple of days. Even so, IP bans can also be placed on entire ranges of IP addresses, so it's possible that even a new IP on the same connection would still be banned. The only real solution then is to get a new ISP, a new internet service, or to move to a new location, all of which seem drastic. (You could run through an IP anonymization service, but those are also frequently already banned, and often impact performance in ways that cause problems for online games.)

Cookies are the most ineffective way to ban. Clear your cookies, and if that's the approach they used, then you'll no longer be banned. Since it's that easy, it's also very rare.

Registry Information works as a banning approach if the site or game includes software that runs on or is installed on your machine. As you might expect, the keys are often obscured so as to make what you've attempted - simply deleting the keys - much more difficult. Uninstalling the program or just looking for keys in the registry that match the game or site name is not likely to work. The good news here is that if this approach is used there are two solutions that are likely to bypass the block: use a different machine, or reformat the original machine. The bad news is that both techniques are somewhat extreme (and of course, not guaranteed.)

Hidden Information can be used as well. Much like registry information, an application could store a secret file somewhere that's not obviously associated with the program that's not deleted when the program is uninstalled. This isn't particularly common, and once again a different or reformatted machine should side-step it.

Required Information is an often overlooked technique, but actually fairly common. This has nothing to do with your computer or technology. You indicated that you paid for some portion of this service, at which point your payment method - be it PayPal, a credit card or something else - does uniquely identify you. You might try using a different payment method. Even your contact email address can be used in this manner.

There are probably other techniques that I'm missing, but you get the idea. For every technique used to ban someone, there are things you can try that may, or may not, circumvent the ban. There just aren't any guarantees.

I still recommend contacting the game's support people and pleading your case there.