Hi Amy!
Any non-alcoholic beverage on the menu you are entitled to. Soda's, juices, pomegranate lemonade (which you'll find at many of the restaurants), are items you can choose. A milkshake is probably considered a dessert, but some places offer smoothies which you can choose as your beverage.
A good example is the 50's Prime Time Cafe' at Hollywood Studios. When we visited there last year, I chose the Coke Float, my wife got a Smoothie, and my daughter got the Bee Bop Drink as our drink selection with our meal.
Each restaurant are slightly different in their offerings. Some restaurants just have the standard Coke products, and others will switch it up with more variety. Unless it was some type of special Signature drink, we were entitled to almost all drink options for the most part. Just be sure to ask them if you're not sure when you look at the menu.
Best wishes to you and the family on your trip this summer!