Most traditional coffee shops serve "melon float," melon soda with vanilla ice cream floating in it.
Actually, my favorite melon beverage is "melon juice," sold mostly in basements of department stores at a counter with a row of blenders. It's kind of like an Orange Julius, only made with honeydew melon. You can also get strawberry and other fruits blended up with whatever milky substance that is.
Japanese beverage vending machines hold endless fascination as soft drink varieties come and go. Aside from the endless variety of coffee beverages and the growing number of tea beverages, I have seen apple soda, mango soda, cardamom soda, and plum wine soda--once each.