I studied in Spain in college and agree that kids will probably like tortilla espanola, the egg and potato omelette. That usually comes with a basket of bread too. Paella does usually have seafood but it also often has chunks of chicken so maybe you could order it without the seafood. You could say, "Paella con pollo y carne" (with chicken and meat) or "Paella sans mariscos" (without seafood).
Another thing to consider is ham and cheese. Spain has the best ham and cheese. Not sure if you are going to be in Madrid or not but they have a chain or restaurants there called "Museo de Jamon" (Museum of Ham). It's a funny place popular with tourists and locals -- they have huge slabs of ham hanging from the ceiling so the kids might get a kick out of that.
You can get ham and cheese sandwiches "Bocadillo de jamon y queso" to go or often at a tapas place they will have an assortment of meats and cheeses on a plate that they serve with slices of bread.
"Bocadillo tortilla espanola" is a sandwich with the omelette in the middle. Sounds weird but it's delicious. We would pack these in our backpacks for lunch when I was a student there.
If you are in Madrid there are lots of little tapas places near the Plaza Mayor. The entrances are on the opposite side of the plaza and they look really small from the outside but they keep going like little caves. They are actually called "cavas" in Spanish. They have lots of short tables and chairs that kids would like to sit at too. It's fun to go and order a pitcher of Sangria and some tapas.
I also second the croissant idea -- especially the chocolate ones! They call them "pan (bread) chocolate." For breakfast you could just stop by a bakery and buy a few things to eat on the go.
If you get sick of eating out with the kids at lunch you could also just by a loaf of bread (pan) at the bakery or grocery store and some meat and cheese, yogurt, fruit and have a picnic. We did that a lot too. The hard part is the stores close during siesta so you have to plan ahead.
The kids would also probably love to go to a chocolateria. It's a place that serves mugs of thick thick hot chocolate. They also have fried dough called "churros" to dip in the chocolate. Spaniards like to go to the chocolateria after a night of partying so they usually open late but if you ask around you could probably find one open in the afternoon/evening.