Gorgeous George's is my favorite place to go with my family. It's located on 78th & Greenwood and serves Middle Eastern food. Chef George is from Nazareth and makes the best lamb shawarma, "George's Grandma's Chicken" (a whole small chicken big enough to share and available Friday-Sunday), fantastic hummus with lamb, great falafel, an incredible fattoush salad, and delicious lentil soup, among other wonderful dishes. The prices for entrees are higher than $14 apiece, but the portions are huge. Most often, my parents and I order two appetizers and two entrees and split them. And we always have leftovers. The restaurant can get a little noisy on Friday and Saturday nights, but other nights are generally quieter.
http://www.gorgeousgeorges.com/dinnermenu.html

emmer & rye on Queen Anne is a great place to go with family as well. It used to be Julia's, so the space couldn't be less romantic and there's little intimacy with the bright lights and other families surrounding you. But, the restaurant is separated into rooms (old house, ya know), so each room's sound is rather self-contained. Also, the menu is extensive and most plates (including apps) are available in small/half portions, for individual ordering, or large portions for sharing. http://www.emmerandrye.com/menu.php

Ray's Cafe works well, especially during the winter and early spring, when fewer people are there. Their menu items for dinner usually hover somewhere in the 11-22 dollar range, so many of the dishes would work within your price range, while others would be a little steeper. The atmosphere is very casual and family-oriented.
http://www.rays.com/cafe/entrees

Another great choice would be Saltoro, on 140th and Greenwood. Their sea salt chicken is delicious and the prices are moderate.