What is the reaction in a child's 'add water and watch the egg hatch' toy?
The egg hatch toys are composed of a superabsorbent polymer. These polymers are able to absorb up to 30 to 60 time their volume in water by forming hydrogen bonds with the water molecules.

Sodium polyacrylate ($\ce{[-CH2-CH(CO2Na){}-]}_n%edit$) is the most common superabsorbent polymer and according to its wikipedia page, is the one used in the egg hatch toys.