Protip: You can't change the IP assigned by the ISP without breaking more than a few laws. I think he's just changing the Private Address Range (192.168.1.???), which does nothing to hide you from a trace.
This.
Dude, I really wish there was something we could do to really help you. Computers aside, this guy is clearly a knob and has no right to do any of this to you. He is an idiot of epic proportions! Honestly, this guy appears to know jack about computers, networking, the internet...hell, anything! If he thinks these computers slow his internet down, then go download the Ubuntu ISO and watch a ton of youtube vids. Then he'll see what slow is...
A) It's not his internet...my advice to you would be to create a list of trusted users and block his laptop from accessing the internet in your house. Or make it so all he can access is a rickroll?
B) If this would make things worse, show your mum some of the posts on here (if you want me to PM you a letter to her, explaining everything, then by all means I'd be happy to).
C) If that's not gonna fly, then all I can suggest is you install some kind of decent AV on your mum's computers. Explain to her about how these forums work. There is no html/javascript embedding, so no virus can be hidden. Show her that when hovering over a link, it shows the URL that you'll be redirected to, so there is no chance of being tricked into clicking something. Also explain that the guy at Staples was just being a businessman, trying to convince her the problem was worse than it actually is, so they can have the business from fixing it.
D) Finally, if none of this works, I'd go for framing the babysitter. Track what he is doing on the computer, and collect a log of all sites he is going on. Bring that back here, and we can sift through the most likely candidates, perhaps finding the culprit? Although it sounds as though this babysitter has a ridiculous amount of authority here.
Oh, also, if the babysitter/[censored] is reading this, I have one thing to say...GTFO of this kid's life! You are a tosser, and you really need to get off your high horse! You are ruining a young lad's life here with your interference, and your fake knowledge of IT is almost laughable! You're bullying a kid because you know you can't take on anyone your own age/maturity, you know that right? And even then, you won't bully him directly, you have to go through his mummy! Just leave him alone, and try not to let the door hit you on the ass on the way out...