Chris is right and I want to reiterate the points he and others have been making:
1) it's the lenses in a system which are far more significant than the camera body
2) in shooting an active young child, whether it be indoors or outdoors, you'll need and want the fastest lens you can afford
3) it is important how a camera body feels in the hand, where the controls are, if everything falls into place naturally for you during use, if the menu system is easy to figure out and navigate, etc.
4) as mentioned, the Nikon D3100 (and the newer D3200) work with AF-S lenses as far as autofocus; other AF lenses will need to be manually focused because this camera does not have an internal screw motor drive. Nikon has issued the basic lenses in AF-S so that is not as much of an issue as it might seem. For instance, there is the very nice, fast 50mm f/1.8, the fast 35mm f/1.8, etc.
5) zoom lenses are very useful but don't overlook the value of a good prime lens, especially a fast one. While the kit lens that comes with the D3100/3200 is pretty decent, it's also not a very fast lens, which would become apparent when shooting in low light conditions.
In other words, think about the kind of shooting situations you're likely to engage in, and where, and what kinds of results you'd like to have. Portraits of your child that you can print and frame? Photos of family trips? Scenic views around your home area?
In addition to considering camera gear, don't forget that you will also need to purchase software for the computer in order to process the images. iPhoto is OK, but it is really lacking in some ways, so many people choose a specific editing program such as Photoshop Elements, Aperture, Lightroom, etc....
As far as the new four-thirds (4/3) or other mirrorless camera systems that are available today, yes, that is another possibility that you should check out. Most of them are not quite as quick as a DSLR but there is minimal shutter lag, certainly far better than in any point-and-shoot. With mirrorless cameras again you have the flexibility of interchangeable lenses, and that is truly valuable. With many of the new cameras you can purchase an adapter and then use older "legacy" lenses. More and more photographers who have long used DSLR systems are adding a mirrorless camera and lens(es) to their gear, as often the lighter weight, smaller camera does nearly as well as the heavier and bulkier DSLR.
I have been shooting for years and have accumulated lenses along the way which I use on my Nikon D3 and D300. I also have recently added the D3200 and the Sony NEX-7 to my gear. Frankly, I love the Sony NEX-7 to death -- it's a marvelous little gem and the image quality I'm getting is quite impressive. However, in shooting wildlife, even the ducks and geese around home, I still prefer to have a DSLR and longer lens(es) in my hands.
Affordability: as others have said, I'll reiterate: put your money into a good lens and don't worry a lot about the camera body itself, as in a few years you'll want something else and yet you'll still have that good lens to put on it.
Good luck!