Unlikely to be Image Stabilisation

@Dan, that's a good question. I suspect that most of us have a problem with blurry images more often that we would like.

It's unlikely that Image Stabilisation is causing this problem. I have certainly never had this problem myself. However, IS reduces camera shake - that probably isn't your problem.

So, what could be causing it?

Shutter speed not fast enough

The rule of thumb often touted is that shutter speed should be faster than 1/focal length in order to avoid camera shake. However, this may not be fast enough for a fast-moving child.

I suggest that a speed of 1/250 or more may be necessary.

Autofocus not working for you

It could be that autofocus is in the wrong mode. Most DSLRs have several autofocus modes. One of these will be "single shot", where the camera focuses when you press the button, and keeps that focus until the photo has been captured. This mode will often result in a blurry subject if the subject is moving towards / away from the camera. Instead, consider a mode which is trying to track the subject. Depending on your camera, this may be called "AI Focus" or "Continuous" mode. (others are welcome to chip in / edit this answer to get the right terms used by different manufactures).

Try stopping down

If you have enough light, you could try a using smaller aperture to give you more depth of field. Obviously, this may slow your shutter speed, so there may be a trade-off to consider here.

Try Sports / Action mode

Finally, if you are using a compact camera, or a DSLR in an auto mode, check which mode it is in. For children, sports or action mode is often best. That should tell the camera to automatically use the highest shutter speed it can, and to prioritise freezing the action.

Hope this helps.