If you were my kid, I would let you have another bunny! hahaha.
My son is only a toddler, but if when he is older and he wants a pet of his own, I'll let him get whatever kind of animal he wants. I've always had cats and dogs. I didn't get a bunny until I moved out and I've had her for 3 years. When my mom comes to house sit, she loves my bun. She gets her out and loves on her and texts me to tell me that she is sooooo soft. haha.
I'm dealing with a similar situation, except its my hubby not my mom. He won't let me get any more chickens. (they're super cheap to have and they pay their own rent with eggs.) He's being a real butt with the chickens, but we've been talking about alpacas. He'll talk about fixing up our barn for them and then he realizes what he's saying and says 'no, what am I talking about? We're not getting any alpacas' Then he calls me a crazy animal person and walks away mumbling to himself about how our animals are immortal. Its funny, its not in a mean tone or anything. He just thinks I'm a nut and doesn't understand why I want more animals.
I would wait to ask her again closer to your birthday, that might get a better response. As a mom and a daughter I don't know what I would do to ask. When I was a teenager we got a golden retriever that I had to beg for, he was 13 years old and my friends were getting divorced. My mom was resistant at first, then I brought him home. She fell in love.
But as a lenient mom when it comes to animals, I would probably cave and jump on the animal band wagon with my son when it comes time for him to get his 'own' animal!