Go find a high turnover job and stick with it. Car washes and busboys, find something unappealing, because people will want to pay you not to have to do these things. The percent of people who look for these jobs is relatively low, everyone wants a nice comfy restaurant server job or a cashier job. Get something that people won't want to do.
EVERYONE IS HIRING, YOU JUST HAVE TO MAKE YOURSELF AVAILABLE.
Aside from that rant that I've been telling all the whiny priveledged 16 year olds I know (sorry), tell him that your computer is too slow, and you need something that will work properly.
You could easily consider (if you've got a spare copy of windows) doing something that will cause your computer to not load windows/it's OS properly, and basically blue screen. Find a way to do this without irreparable damage, or find a way to easily get it back working a few months after you get the pc, and say you were tinkering with it for that long, and got it working again.
Consider talking openly about something that you are getting into, such as game development or music, or video editing or webpage design. Something that your laptop is going to struggle with.
I'm not saying these are all morally straight ideas, but they would technically get the job done if they are implemented correctly. I would advise the last suggestion, personally, about content creation needs, but please exercise extreme caution and make sure that you are aware of all the variables before setting down a road that will pave you into a neighborhood you don't want to or can't get out of.
Consider talking to your dad and telling him that you have no money to get a decent computer and you want to learn about or study computers, maybe go get A++ certified (comptia) and tell him you'd need one for IT work.
Consider living with your mum if he's not willing to get you a pc after getting himself some cars and trucks and whatevers.