One thing that children often seem to find incredibly fun is balancing toys — things that stand up when it seems like they should fall down — things like this toy "Balance Bird":
(I bought this one at the gift store of the Museum of Osteology in Oklahoma City — a fantastic museum, but that's another subject. You can find lots of these things if you go to Amazon and search for "balance bird.")
Not only do they not fall down, but if you bump them or tip them, they sway and swing and come back to where they started. They actually behave a lot more like something hanging than something standing. If you inspect the bird closely, you can discover why. The bird is mostly lightweight plastic, but there are heavy metal weights stored in the wing tips. And if you look closely, you see that the weights are actually lower than the point of support, i.e. the tip of the beak.
If you want to make a similar toy from household objects, it is not that hard to do. The important thing is to keep most of the weight centered underneath the point of support, or in scientific terms, to have the center of gravity below the point of support. One way to do this is to jam two forks into a piece of clay or a piece of potato or something, and then insert a toothpick in the center.
The toy is to be suspended from the toothpick, somewhere in the middle. Notice that the forks are tilted backwards so that the tips of the forks are well behind the toothpick, thus balancing the toy front-to-back. You might have to experiment a little with the angle of the forks, and the weight of the lump, to find a happy balance. Also notice that the toothpick is off-center. It is above the centerline of the toy. This is necessary to place the point of support over the center of gravity, rather than under it. It is necessary for the same reason that the bird's wingtips were below its beak. But once you have found a happy balance, you should be able to hang or stand the top from a point somewhere in the middle of the toothpick.