What should I do to convince her?

sorry MOMs rule not kids.
References :

Mom's house, mom's rules. Do you have the money to pay for a vet if the dog got sick? Do you have the time to properly socialize and train the puppy?

Wait until your older then show your parents how responsible you are and ask again. Begging and pestering won't get you what you want.
References :

If your mom says no then it's no. You just have to accept it.

Today must set a record for the number of times this question has been asked ~
References :

If your mom says no then it's no. Sorry kid. You shouldn't have gotten attached to a dog that you can't have. Under your parent's roof it's their rules. You'll have to wait until you move out to get a puppy.
References :

that was wat happened to me too… well make sure that you also have enough money for its shots cuz that is super important if u want ur dog to survive. well mention to ur mom that poodles don't shed… they don't their fur just doesns't shed its pretty cool.. cuz most of the time thats wat parents worry about is the hair everywhere… also tell her that you'll clean up every bit that puppy does and that it won't do that stuff cuz you will b training it and taking it on walks.

- i wish you the best…. tell me how it turns out
References :

i think thjat u should make ur friend feel guilty and say stuff like "oh but ur my bf" and that "if i had a puppy i would give it to u for free"
References :

What do you do? You grow up and stop whinning like a spoiled little child and accept that your mother has said NO, means NO and will not appreciate being pestered over something she has given you an answer to.

Didn't realize we were dealing with quantum physics here.
References :

Sorry, I have to agree with the rest of the folks. Owning a dog is very expensive if you do what you are supposed to do. Food and other stuff, is not even the beginning. A dog needs to be neutered. It needs shots, it needs flea preventative, it needs heartworm preventative - not just once but always. It needs to go to obedience school. It needs a license. And then, if the puppy has an accident or gets sick it needs more medical attention - that gets extremely expensive.

Puppies poop and pee all over until they are housebroken. Then they chew. They chew on everything. They have behavior issues. How much experience have you had with all of these? It is not hit and miss. You have to know how to handle these situations, you need experience. Doing the wrong thing would only harm the puppy.

Why do you want a puppy? How much do you actually know about owning a dog? Are you really into dogs or just this puppy because it so cute? If you are only into this puppy because she is so cute, then give it up. Puppies always grow up. They don't stay little and cute for long. That's when people decide they are tired of them and dump them at a shelter, where most are put to sleep. God knows there will always be another puppy looking for a good home.

If you really like dogs then you should ask your mom if you can volunteer at a local shelter. That way you help some dogs in need, gain some experience and responsibility, while proving to mom that you are serious.about getting a puppy. Or you may discover that it really is too much work and responsibility.
References :
Experience.