How do I convince my mom to let me dye my hair?

Honestly, I think your mom is just trying to look out for your hair. But the only thing that anyone could really suggest to you is that you could say that it's your hair and if something happens to it, then it would be your fault and you would have to live with the consequences. Other than that, just have a calm conversation with your mom and she might let you if you're nice about it.