So I live at home with my mother and brother. I pay rent, comparatively not much but the amount was agreed upon. The other day I got home from work to find out she had not only gone through my bedroom without my permission, she had also found bank statements at the bottom of my draw and decided to look through them, again without permission. We have in fact had multiple arguements where she wanted me to show her my bank account and I had refused. Is there any legal action I can pursue to dissuade her from doing this again in the future?
Make a written contract. Lay out in it things like access to your room and financial situation.
By having a contract, any action against it is a breach of this contract and could have consequences laid out in said contract as long as these are reasonable.
Since you are paying rent, you are a tenant. One of the things that a tenant is normally entitled to is the "quiet enjoyment" of the rented premises, which means that the landlord cannot enter without invitation except in emergencies or agreed circumstances. In theory you could sue for damages for that violation of the tenancy. But you might not be able to show any specific damages, and even if you could I doubt that you would want to make a legal case out of this. A locked file cabinet or cash box for those records that you wish to keep private might avoid future problems.
At present, it’s her house so she can do what she likes
The arrangement between you by which you pay rent is not a contract - partly because the terms are too vague but mainly because arrangements between family members are assumed not to be contracts unless the intention of both parties to treat it as one is manifestly clear.
I agree that your mother has overstepped social boundaries but she has not crossed any legal ones.