My friend J sent me a link to this gyrobike (http://www.thegyrobike.com/index.php). From what I can tell, it’s a flywheel that you put in the front wheel of a bike. The site claims that this will help kids learn how to ride a bike.
So, what do I think? Clearly, this is a real product, but I am not so sure how effective it would be. As I mentioned in a previous post, it’s not really about angular momentum that prevents you from falling over. Well, actually I didn’t really say that. I referred to an excellent article on the physics of bicycles, David E. H. Jones and published in Physics Today in 1970 (The Stability of the Bicycle 23 p34-40.) (you can find a link to that article at The Physics and Bicycle page) Jones tested the idea that angular momentum keeps a bike up. To do this, he created a bike with an extra counter rotating wheel. This made the total angular momentum of the bike zero (the counter rotating wheels had opposite angular momentum). From this, he found the bike was easily “ridable”.

So, if reducing the angular momentum doesn’t make the bike more difficult to ride, I would imagine that increasing it wouldn’t make that much difference either. One thing he did find was that the zero angular momentum bike didn’t ride well as a ghost-rider bike. It just fell over. So, maybe the gyroscopic action of the wheel may add to stability in the case of a low mass bike. However, when a rider is on there it might not make much of a difference.
Here is a video of the Gyrobike people testing the ghost-riderability of the gyrobike.
If you want to learn how to ride a bike, I recommend just removing the pedals. This allows a child to learn how steering and balance on the bike works without having to worry about pedaling. This is what other’s recommend.
Further, if you want to make a super stable bike, I have a better idea:

This biker uses a long pole to help balance. The pole makes the rider have an object with an extremely large moment of inertia. You could do this with a kids bike also, but it might be difficult to pass through narrow openings.
Let me just emphasize that I don’t know the gyrobike doesn’t make a difference, I am just suspecting it doesn’t make a difference. I totally could be wrong, it happens all the time. Maybe the gyrobike does work as a placebo effect.