Children under 11 can travel for free on all national rail services in London.
Previously, there was something of a north-south divide (isn't there always): 5-10 year olds got free travel on most routes north of the Thames, whereas operating companies like Southeastern, Southern and South West Trains charged child fares. That disparity is now wiped out — but beware the small print.
Children will need a Zip Oyster photocard to qualify for free travel — it's not like buses, where children can just rock up and get on. Your child will also need a photocard even if they're travelling with you, unlike the tube where they travel free with an adult. The card costs £10 but it's obviously worth that initial outlay. Transport for London has even put together this video explaining how to apply for a Zip Oyster card.
Thanks to friend of Londonist, Clare Griffiths, for the tip-off and photo.