Without hesitation, I'd go with GameMaker along with the book The Game Maker's Apprentice, and followed by The Game Maker's Companion. The first book includes an older version of the GameMaker software which is all that will be needed to complete the exercises. If your child likes the process, move onto the second book which covers more advanced concepts. Those books, along with either GameMaker 8.1 or GameMaker: Studio should your child want to move onto more current versions, will all fit within your $100 budget, and it will only cost you $20 or so to get started.
The books are excellent learning tools and the GameMaker software itself was originally created by co-author and Utrecht University professor Mark Overmars to teach programming. It's a great way to get ones feet wet and very good games can be created with it if one is willing to put in the effort. If you child wants to move on to more popular languages, GameMaker will provide them an excellent foundation for learning them.