Idea's To Cinvince My Mum To Let Me Build New PC. Help!

Go out, get a paper round or similar job, stick with it and after a while,
approach her with a 'Xmas present/birthday' deal, you save xxx up from your wages, and your parents match it,
I can't say much in the way of persuasion because I don't know your families situation,
maybe theres no spare cash around and she won't see a new pc for you as a priority over paying bills etc
take initiative, show her you want this so much that your willing to work for it and you might find her a bit more willing to come to a solution,
Moto

another good approach (what i did besides what Moto was talking about) is show interest into a career with computer. I pretty much told her what i wanted to work with IT when i grow up and i worked hard and bought bits and pieces of my first computer.. then when xmas/bdays came along, i ask for the rest of the parts to finish my build.. takes time, but hey, it works when they see you really wanting to learn how computers work.

brood a lot because all your friends got better computers

sadly, paper routes are kind of a thing of the past...kids don't get them anymore.

I suggest being slightly subtle: get books on building computers out of the library, and make a point of actually READING them in your parents' presence. show them you're seriously interested in building your own PCs as a hobby or career choice.

As long as people are alive then theres work to be done,
and a man with the mind can find himself (or herself) someone willing to pay money so as they aren't the one working,
Paper rounds may be on the slide, but I reckon theres plenty of other options Op could find to bring the money in, and show Mum that he's serious
but plus points for the learning aspect as well of course  
Moto

Thanks everyone for your replies, I think it has worked. :-)