Is there a reason your mom doesn't want you to have a snake?
If she's afraid of them, then JakeTheSnake's suggestion would be the most appropriate way to solve the problem. But some people never change, and it may even traumatize her, so don't push her too far, just close enough so it can perk her interest. If she's still thoroughly terrified of them, then just back off for a while I guess and try again later. If this is your case, then you're gonna have some trouble.
My mom still thinks snakes are the most horrible things in the world, but I keep mine out of her sight in my room (also keeps her from nosing around my room too

).
I guess the best thing to do would be to ask your mom what any other kid would ask "why?", and refuse to settle down if she just says no, and tell her you want a reasonable and logical answer. Once you have that answer be it "they're too expensive", or "they're dangerous", or whatever it may be, your next step is to prove her wrong. Can't really help you there since you have to rely on your own wits and since you know your mom best.
The best and easiest thing would be to just wait till you're old enough to have a bit more freedom, like when your out in college or something. But then again, there are those who can't wait and must have things when they want it and what not.
Good luck kiddo, I remember I used to be in your same exact shoes 5 years ago when I wanted a rifle, and I showed my dad what I could do with my friend's 22 rimfire rifle(one hole groups from 25 meters away), and bang, I got a my first rifle on my B-day.
-13mur 6