M.,
I would go with hardwood floors because they appeal to more people when you go to sell your house.
Also, carpeting off gasses formaldehyde for a really long time and it's not healthy for anyone to be breathing, but its especially bad for children because they take more breaths per minute than adults.
If your children have indoor or outdoor allergies, wood floors can be washed and the allergens removed. Carpets cannot be completely cleaned and still harbor things like dust mites even after a cleaning.
We have hardwood floors in most of our house and only one room that has carpeting. Our son loves to play on the hardwood floors and not on the carpeting. But, it may be a personal preference.
I would stay away from laminate flooring just because the lifespan is, on average, 10 years or so. If you want a floor to last for awhile, then I would recommend bamboo, cork, or a solid wood floor. However, if you like the idea of bamboo and cork flooring, it's worth the time to ask the flooring company how the plants are glued together into a plank. Some companies use a formaldehyde based glue and others do not.
Good luck making a decision! I'm sure the room will look beautiful once you are finished.
C. J.