I went to a private school too up until 8th grade. I agree with the first answer--just because it's a private school doesn't mean stuff that happens at public school doesn't happen there. In fact, I think the kids in private school are worse than kids in public school. They're snobby, stuck up, and gossip worse than anyone I've ever seen.
I'm homeschooled now, by the way. I absolutely love it.
You say that go out with friends so you won't worry about socialization, but you should probably know that if you start homeschooling your friends might not want to be around you anymore (unless they're really good close friends). I mean, the only thing teenagers really have in common is school and if you don't go to the same school (or kind of school) you kind of get disconnected.
The good news is that if you become homeschooled you can make TONS of new friends--not just friends that are in your class. And, you'll have so much free time that you'll be able to spend as much time as you want with your friends. You don't have to worry about all the crap that goes on in regular schools if you're homeschooled. Chances are the new friends you'll make won't care what brand your clothes are, what kind of music you listen to, or what guy you think is hot. They'll like you for you who are.
If you want to convince your mom to homeschool you, you'll need to come up with a good list of reasons why. Not just "I can't stand regular school anymore.." Sit down and think hard about why you want to be homeschooled and make a list.
Then, I would suggest researching different options. Whether you want to have your mom teach you, or if you want to hire a tutor, or if you want to do online schooling. If you show your mom that you've researched this it will show her that you're serious.
Lastly, I would show her all the benefits of homeschooling. You'll be able to improve your grades because you'll have one on one interaction with the teacher. You'll be able to learn at your own pace and you'll be able to find a curriculum that fits you best.
It really doesn't take that much work to homeschool. I know kids whose parents both work and they are still homeschooled. If your mom works, she can set up your assignments in the morning before she leaves and you can work on them throughout the day. If there's something that you need to be taught, she can teach you when she gets home. Or, you can even get dvds that have a teacher who teaches you as you follow along in your book so that's an option too.
Anyway, I hope that helped. Good luck..I hope she lets you :-)