I hope you mean that your Mum wants the dog, and given that opportunity you think it will be nice for your child to be a little bit involved.
If your mum wants the company a dog, and the dog is to be hers, live with her, and she'll be the one primarily responsible for it's care then it's her decision.
Has your mum any experience of keeping dogs?
Does she really want a puppy and all the work that involves vs. an older dog?
You need to draw up a list of who wants what from a puppy/dog, and a list of what between you you can offer to a puppy/dog. The limiting factors that determine whether it will work and which breed may be suitable will involve the facilities available (garden, size of house, general health of your mum, etc.) and how much everyone can realistically contribute to the needs of the dog (guaranteed minimum exercise hours, food and insurance bills, etc.). You must also consider whether the limitations of dog ownership are something you can accommodate (restrict holidays to "dog friendly" or arrange care/boarding, etc.).
Expecting a child under 5 to be involved with looking after or training a puppy is unrealistic, bordering on fantasy. You'd do well to wait several years if that's what you want to get from this.