You can't get a three year old a dog. I even wonder whether it is a good idea for me to pay for my son's pup after he turns 21
However, if YOU want a dog, then it is possible with a youngster, though I would tend to wait until the child is about 5 and can be reasoned with unless the child has grown up with a dog in the house.
Children can mither dogs. If your three year old decides to constantly bother the dog, it isn't going to be ideal for the poor dog. At around five, they can learn basic training and how to act around dogs and understand why rules around the dogs are important.
And remember puppies 'mouth'. It is normal and natural for the pup, but needs to be taught out. Tiny, sharp little teeth, no bite inhibition and a three year old mad keen on puppies probably isn't a good mix.
That said - if
you really want a puppy, the Cavalier King Charles Spaniel generally has a good temperament and is good for the first time owner. But ensure you buy from a good breeder who fully health tests their dogs and knows their lines - they are prone to a painful and distressing brain disorder in addition to patella and eye disorders.
Alternatively, a well bred Staffordshire Bull Terrier - again from health tested stock and breed for temperament - is generally a good choice around children. Though may be too bouncy for a three year old.
And think of the cost - between £500 and £800 for a well bred dog, £100-£200 for equipment, £200 for a year's insurance, £50 jabs, £100-£250 for spay/neuter - that is £1500 in the first year!!!
And do you have time to go to training/socialisation classes? As a first time owner, they are essential.
Another option is an adult rescue dog that has come from a family with young children. Many child friendly dogs end up in rescue due to relationship break down, financial difficulties or having to move into rented accommodation where dogs aren't allowed. A good rescue will assess the dog's temperament - preferably in foster care - and match suitable dogs to owners. However, most (but not all) rescues won't home to a family with a child under five.