My 8 year old isn't what I would call a picky eater, but she's not always keen on trying new things. What offerings can I expect for kids in Spanish restaurants?

the wonderful thing about most spanish restuarants/cafes, bars.. is that they have a hot grill in kitchen and can whip up anything SIMPLE to your desire.
grilled chicken breast.. homemade french fries everywhere.. pasta..

you will need to learn the words of how you do/don't want something cooked.. "poco aceite" sin ajo" a la plancha.. sin sal.. (litle oil, no garlic, grilled, no salt, etc)

white rice.. grilled ham and cheese sandwiches, salads.. and the soups and omelettes are plentiful even if it is not on menu, they will make you a french omelette (eggs only) anywhere/anytime the grill is on.

Don't forget -- the best "french" fries on the planet! Fried in olive oil, so they're healthy, right??

If your daughter likes hard salami at all (my kids gobble it up like candy, unfortunately) she might like "salchichon" which is similar. The chorizo will probably be too strong (and chewy) for her. Same goes for the jamon serrano. "Regular" ham is called Jamon York.

Possibly the ubiquitous Tortilla Espanola (potato/egg omelette)?

You should also introduce her to churros and chocolate -- really thick, yummy hot chocolate. She'll feel like a true spaniard while you get your caffeine for the day ;-).

I lived in Spain when I was a young girl and loved going out to eat. Spanish food is wonderful.

My favorite thing was to order entremeses (appetizers) and eat them all myself. Gave me lots of stuff to choose from and was fun. I gave the mussels to my Mom lol.

I used to have calamari, tortilla, pollo any way, and loved the jamon sandwiches, shrimp.

I sooo miss the papas and pipas. (french fries and sunflower seeds sold on the street corner in gray paper cones)

You can always fall back on spaghetti which is on most menus. In Madrid, we encountered a chain called Vips which has a very kid friendly menu and the food was good. (Ours was just out of the Retiro park near Starbucks).