Maybe you could print out pics of hamsters and info about them, and show them to your mom? That's how my mom let me have my rabbit when I was ten. Hamsters are excellent pets.

My mom hates rats (and she says gerbils look like rats), but she allowed me to have my gerbil because she knew it was what I really wanted and that I am responsible enough to take care of him. Why don't you show your mom how responsible you are, like cleaning your room, doing extra chores around the house, etc. Even if it is not a responsibility issue, it may do the trick.

That's all I can think of, besides maybe printing out these replies and putting them somewhere where you know your mom will read them?