I'd suggest
http://www.abeno.co.uk/
near the British Museum and very kid friendly, eating here is potentially an experience as the food is made at the table.
(Abeno Too is near Leicester Square, another convenient location)

Also, when you're checking out the houses of parliament (Big Ben) there's this charming place for lunch:
http://www.gardenmuseum.org.uk/visit...rden_cafe.html
lovely and atmospheric with decent food at a decent price

And if you're near Oxford Circus there's:
http://www.scandikitchen.co.uk/menu.html
This place is very popular with those working near by so if you can avoid the peak time of around 1pm it will be less of a squeeze.