How does one go about convincing their stubborn mother to get a puppy?

You grow up and get your own place. Sounds like your mum is living in a menagerie already, you will only sound like a spoilt brat if you start splitting hairs over who owns which exact animal. Most dogs and cats are FAMILY pets, and your family already has enough to go round. I pity your poor mum, if the dogs and cats you have now aren't housetrained why would she want another one?!