Even in literature and art, no man who bothers about originality will ever be original: whereas if you simply try to tell the truth (without caring twopence how often it has been told before) you will, nine times out of ten, become original without ever having noticed it.
-- C. S. Lewis
Game-related ideas come from many sources and can be in many shapes and forms. At some point, ideas coalesce into something that can become a game. Usually there is some immediate stimulus, some spark, involved.
I want to discuss the kinds of sparks that are common to new games, that is, where and how do games originate?
Ideas for new games usually come from one or a combination of several aspects of games. These are:
theme (story, title, image)
a particular game, a game system, or a genre
components (mostly, non-electronic games)
The occult-looking lines in the diagram are meant to indicate that a game may have more than one kind of origin.
Let's describe each in turn. In most cases, only the designer will know the origin of a game, so our examples will be limited.
Theme: Story, Title, Image
The theme is some set of circumstances, usually a story, that can affect the game's mechanics, appearance, and gameplay. It may be as simple as a title or an image, in the imagination or in a tangible form, of some event or activity.
Most Star Wars games have a theme deriving from the original Star Wars films (1977-1983), Civilization has a theme of the rise and development of civilization. Age of Empires is a more consistently military approach to the same idea. Britannia is a board game where the theme is a thousand years of British history (after all, "story" is integral to "history").
In general, any history, real or imagined, as the Star Wars history is imagined, is a theme. There are many board games based on (that is, borrowing the theme of) video games, and vice versa. For example, Civilization the computer game, though not directly derived from Civilization the board game, is certainly related to it, while Starcraft: The Board Game is clearly derived from the computer game.
Sometimes the story is very simple, as in the board game Dragon Rage: attacks by monsters, sometimes dragons, on a city. The title alone helps characterize the game. At least one game's theme comes from a story that was written to support a set of commercial miniature figures (Valley of the Four Winds).
Many European-style board games have tacked-on themes that don't affect gameplay or mechanics at all, though they affect the appearance of the game. Despite the name, the gameplay in the board game Ming Dynasty has exactly nothing to do with China, though the artwork is vaguely Chinese. Some stories are merely excuses to blow things up, as in many shooters. In these cases the game probably originated somewhere other than through the theme.
Many AAA video games aim at "dream-fulfillment", a subcategory of theme/story that some might list as a separate kind of origin. What kind of hero, or "star", or expert, or even god, do you want the player to "become" through your game?