An animal of any kind (even one as small as a fish or a hamster) is not a light, last minute purchase. Bringing a new life into the house should be well thought out and discussed with the entire family.
Holiday pets often get ignored in the holiday rush. Christmas morning is filled with so many presents, lots of food, family and relatives coming over...then there's New Years in a few days. You think it's stressful on you? Thank about what a pet who's never been in your house before would be thinking. A new pet needs lots of quiet and calm. A new puppy or kitten needs to watched constantly and settle into a routine so they can become a happy member of the family. This is impossible to accomplish on Christmas. The new pet will just end up confused and scared.
You should never pick an actual pet for another person, even a child. Bring the child along to pick out the animal and let it be a family event. All animals (even hamsters and fish) have distinct personalities and letting your entire family help with the choice makes the animal more special to them. Besides, don't you want to see how the puppy interacts with your entire family? That great puppy you pick out for your son might not like kids. Your son might decide the puppy you like plays too rough. Your kids may decide they'd rather have a cat!
New PuppiesAlmost every child asks Santa for one, however a dog is MAJOR purchase and a new puppy needs lots of attention and care. With the hustle and bustle of the Christmas/New Year holiday, the puppy probably won't get the attention it needs. That's not even taking into account all the ribbon, trees, rich Christmas foods, chocolate and other dangers the puppy could unintentionally get in while your family is busy with their other gifts.
Alternate ideas: Give the kids a stuffed puppy and tell them the new puppy is coming. Wrap a puppy bowl, collar, crate and other puppy supplies with a "certificate" to get a puppy at a later date. All of this stuff should be set up and ready for the puppy when it comes home anyway. This way, you and your family can set it up while you tell them about the responsibility of a new dog. Another great idea is a few books on puppy care (especially if you have an older child).
New KittensKittens don't take quite as much attention as puppies but they can still get into a lot of trouble at Christmas. Kittens are notorious for swallowing tinsel and ribbon and getting lots of stomach problems. Small kittens scare easily and the safest retreat will probably be up the tree which can be dangerous.
Alternate ideas: Cat care kits, litter boxes, cat toys, books on kitten care. The litter box and a bed for kitty should be in place before he gets to his new house. You and the kids can decide where to put it.