We visit DL pretty frequently and our DS is now almost 16 months old. During the last three trips (ages 12 mo, 14 mo, and 15 mo) he has been to several different Disney restaurants.
I would say that his favorite two meals were the Lilo and Stitch breakfast (PPH) and the Goofy's kitchen brunch (DLH). Lots of kid friendly foods (but delicious choices for mom and dad too) and he LOVES characters at this point and thus he delighted in everyone who stopped by the table to visit. Minnie has a similiar breakfast in the parks, but we did not have a good experience there compared to the other two--quality of food was lesser and we had to ask to have characters come to our table after a 45 minute wait, and then still had to wait another 30 minutes or so!)
We also really enjoy the Storyteller's dinner buffet at GCH. There are no characters there, but once again the food is pretty good and DS enjoys what we share with him.
For most counter service places we tend to feed DS from the snacks that I have brought along rather than ordering the kids meals for him. I pack string cheese, yogurt sticks, goldfish crackers and all the other makings of a power pack meal at a fraction of the cost. The one exception is at Whitewater Snacks in the GCH, which is probably my favorite CS in the resort (except for maybe the Pacific Wharf soup and salads). There we order a fruit cup, which is absolutely huge and DS can eat off it for 2-3 meals. The fruit is always very tasty and usually has a pretty good variety of melons, grapes, pineapple and strawberries that the whole family enjoys.
Anyhow, the only places I would recommend skipping in the parks are Wine Country Trattoria and Blue Bayou as they are not really toddler friendly on their menu nor in their atmosphere. Otherwise, pick what you like and pack some snacks for the little one, and have fun!