Just speaking from my own experiance here, it took me a while, but now I know the key
The golden rule is BE PERSISTANT. This doesn't mean the old whinge and whine, that only shows that you're immature and obviously don't deserve a responsibility.
Just keep saying how much you love snakes and how cool and easy to care for they are etc.
The next rule is BE EDUCATED. Do heaps of research on the species that you want, then tell your Mum a few cool things about them.
The next rule is LET HER ADJUST. If I were a mother and my kid wanted a snake... ok probably not the best example...umm.... lets just say a Cockroach. If my kid wanted a cockroach, I would be horrified, and I decided I didn't want to get in the way of their hobby, I would need to adjust to the idea. Just try and put yourself in your mother's shoes.
Take her to meet some snakes etc. and let her adjust to the idea of you keeping one.
The next rule is PATIENCE. Be very very patient with her, it's probably a big thing for her to adjust to, if she's as frightened as you say.
The final rule is TAKE RESPONSIBILITY. When my mother first said I could have a snake, I had to promise to keep it in my room away from her and that she would never even have to see it.
If all else fails, you might have to wait a couple of months then try again.
Cheers,
-Jordo