How do I convince my mother to let me be home-school? I've told her my reasons, social anxiety and it being too easy or hard depends on the subject. I showed her some websites about it and the pros and cons. Any ideas?

Unless you have actually gone to a psychiatrist and been diagnosed ... You can't claim to have "social anxiety".  Doing so is a disservice to all those who actually have the disease.

Everyone has some degree of social anxiety. It is just some are better at hiding it than others. Is middle school/high school tough? Sure. The kids haven't figured out how to properly learn to socially interact with one another. And where do they learn to do that? In high school. By bailing out and hiding at home isn't doing you any favors. It is eliminating the chance for you to learn to deal with people, both good and bad.

So unless, as Walt says, you have not been professionally diagnosed with social anxiety there is no reason for you to claim that as an excuse. Why not ask your mom to help you find an activity that can help you build your confidence instead. For example, a lot of people who have trouble interacting with others join the drama club. They figure they can learn acting skills so they can just "act" confident when they aren't. And don't worry, that does not mean you need to get on stage in front of everyone. There are lots of people needed behind the scenes to make a play run smoothly.

If that isn't your thing look for another club or activity that works for you. It doesn't even need to be in the school. It can be community based. Like a hobby class like cooking or photography. At least there you will be who have something in common with you and it will be easier to learn how to connect with them.

I agree with Aux Arcs.  You need to confront your social anxiety and have it treated.  It can hold you back in life.  If you can't manage high school, how will you deal with the world of work or college?