How picky are your kids with food? Most places are suitable for kids those ages. I have nephews 8 and 11 and take them to most places I would otherwise go as an adult.
For example, if your kids like seafood and fish, I'd suggest La Paradeta. It's a no-nonsense unassuming and inexpensive sort of seafood fast-food sort of place. Imagine McDonalds but healthy :) http://infocatalonia.eu/w/gL2bB and also http://www.spottedbylocals.com/barcelona/la-paradeta/ Incidentally they have three restaurants in town, one located in El Born area -which you're bound to visit anyway- and another one near Sagrada Família.
There's this magazine, timeout.cat which is sort of an 'unofficial' bible for us locals when it comes to going out in the city: restaurants, entertainment... You might want to have a look there, there are plenty of ideas and suggestions. Specifically for children: http://www.timeout.com/barcelona/families/child-friendly-places-to-eat although I'd say that yours are probably old enough for plenty more sites where any of us would go too. I'd suggest to focus instead on the "restaurants & cafes" generic page: http://www.timeout.com/barcelona/restaurants
Whatever you choose, please do avoid the bars "in" Les Rambles as most of them are purely and simply tourist traps: awful food and very expensive.