Transport for London offer free travel for Children who are under 15
This is available even if you live overseas BUT you do have to apply at least 4 weeks in advance of your arrival date in London
For Children under 11 - the Zip Oystercard gives them free travel on the whole system
For Children 11 to 15 - the Zip Oystercard give them free travel on the buss and trams - the card allows them to pay a childrens fare if they use the underground
For more information and to make an application - follow this link