American pit bull terrier owner here. I've owned them for going on 10 years now. Well, my family has anyway.
To the person who said pit bull puppies are the same as any other puppy: wrong, wrong, wrong. LOL You've obviously never had a pit bull puppy that was under three months old.
When my sister took in Lilo (our American pit bull terrier) she was pregnant. She had the puppies (they looked A LOT like your puppy; so, I'm willing to bet that dog is purebred) a few weeks later. They were CRYBABIES if I've ever seen any. So, if she hasn't started crying yet; I'm warning you ahead of time. She might. Lilo's puppies spent ALL their time crying and crying and crying. And crying some more. They were the biggest babies. And, not much changes by the time they grow up.
That's why pit bulls make horrible outdoor dogs.
However, just like any other puppy, they need their nourishment. That puppy looks like she's about 4 1/2-5 weeks old. They can be weaned that young; but, it's very risky. My mom had to wean a pomeranian that young because her mom wouldn't feed her- and, when she grew up, the pomeranian was only 1 lb and had never really "developed" right. Some canned food is a start. That's how a lot of people wean puppies. With canned food or moist kibble. I'd get some puppy formula from your closest pet supermarket. I'd say Wal Mart, but I've heard bad things about their puppy formula. However, if that's your only option, it's better than nothing. There was one puppy that Lilo wouldn't nurse- my sister got some puppy formula and an eye drop to feed the puppy. A baby bottle with a SMALL hole works fine, too. I think they do make bottles for puppies and kittens, too. Though, I'm not sure where to find them.
Since you have small dogs; if you have no experience with pit bulls; I wouldn't recommend you keeping the dog for yourself. While, if raised around dogs, they can be friendly to other dogs. (Lilo especially loves small dogs) However, if a fight does start, they are usually the one to finish it. (though, rarely the one to start it) And, to an unexperienced owner- this can be quite the challenge. Is there a no-kill shelter nearby that takes pit bulls? A pit bull rescue in your area maybe? Just in case your son doesn't want a dog? If he does, though, I say- go for it.
PS To the people who said, it's all how you raise them. It isn't
just how you raise them. Genetics and breeding DO play a big part in a dogs temperment. IE a poorly bred pit bull could show problems in the future. As could a poorly bred yorkshire terrier. Though, raising them properly does help- sometimes, you can't deny that some things
can be inherited. Ever read Marley and Me? Remember that bound of energy that ran past them when they met the puppies? Remember how Marley grew up to be JUST like that? Even though they tried training him as good as they could? Any breed of dog could be the same way.