Just say "Mom, I've been doing a lot of research, and would really like to keep a betta fish, I know you have some concerns, and I'd love to hear them, but I really think that a fish would be a very healthy and rewarding thing for me, and I'm very prepared to care for him." Also, make sure you pick a time when she is free to talk, not when she's grumpy, tired or running off somewhere. Perhaps clean your room without being asked and then sit down with her?
You can even type up all the information in one place so you have it for when you are talking to your mom. Moms love to talk to their kids, believe me! She would be really impressed and touched if you sat down with her over a cup of tea and discussed getting a betta.