Just as there are many types of games, there are many ways to approach the Game Design merit badge. To help you get started, think about what you like to do.
Do you like making things? If so, consider designing a board game for which you would need to make the game board and game pieces.
Do you like playing sports? How about coming up with a new type of sports game? Perhaps you might somehow combine the rules of two different games — such as baseball and capture the flag — into brand new game.
Do you want to delve into programming? Download a free-to-use programming language like Alice, or a purchased one like Game Maker.
Do you like to tell stories? Write a new scenario for an existing role-playing game.
Whatever approach you take, be sure to review the merit badge requirements to make sure that what you want to do satisfies those requirements, and then talk to a merit badge counselor before starting any real work.