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?
The non-electronic version of Dungeons & Dragons can be seen as a game originating in mechanics. There were many fantasy games, but the role-playing mechanic, which more or less began the role-playing game genre, is the defining characteristic of the game.
Dance Dance Revolution and Guitar Hero can be thought of as games that likely originated in mechanics. In a postmortem of the latter game published in Game Developer magazine (February 2006), two of the developers from Harmonix, Greg LoPiccolo and Daniel Sussman, said Guitar Hero was really designed around letting the player feel like a rock star, rather than a game designed around a new controller. The peripheral device came about as a result of giving this rock star feel to the player.
In both Guitar Hero and Dance Dance Revolution, we have an unusual mechanic, the flow of directions that the players follow to make certain physical moves.
A Particular Game, a Game System, or Genre
A "system" is a case where a set of mechanics has become so well-known that games are made using most or all of the set. Many historical board games begin with a system, such as "block games" (Hammer of the Scots), "card-driven games" (We the People), Risk-like games (Risk Godstorm), Britannia-like games (Italia, China: The Middle Kingdom), and "committed intent" games. Video games, apart from obvious sequels, very often adapt a system, and video game genres themselves tend to involve challenges to players that are common to most games of the genre.
Many video games originate with a genre. "We want to make a real-time strategy game," or "let's make a shooter." There are genres in non-electronic gaming, but they tend to be broader (RPGs, collectible card games, war games). The genres in video games are quite specific, tending sometimes to straightjacket the designers' efforts.
Often the genre goes hand-in-hand with a theme or system. Battle for Middle Earth is a The Lord of the Rings themed RTS. LOTR Trilogy Risk is a Lord of the Rings themed Risk-like game.
In the end, many games derive directly from specific other games. In the video game world, the "safe" way to go is to design a game that is much like an existing successful game, but just enough different to be unique and to be perceived as an improvement. While derivation from another game is probably the most common method of origin, it is also probably the least successful, because too many resulting games suffer badly from being "too derivative."
Components (primarily non-electronic games)
On the non-electronic side, components can be at the origin of a game. In my own experience, Law & Chaos originated because I wanted to make a game using the "jewel-like" glass beads that have become popular for plant displays, and another game originated in a desire to use stackable plastic pieces from an educational supply house.
A component could be a special controller, such as one allowing the video game player to "drive a car" in a natural way. It's possible that Dance Dance Revolution was derived from components. I don't know whether the mechanic or the component came first -- or maybe they came together.
I'm going to discuss constraints at some length, as this is where the greatest variety and the greatest limitations can come from.
Are you a person who works better when faced with a deadline? I believe that many people do better work when faced with constraints, whether deadlines or something else. This is particularly true in art, but likely true in most walks of life.
In effect, everything a designer does is considered working within constraints. The answer to the question "Who is the audience" provides constraints. If your audience is preschool children, you can't design a game that requires a lot of math (or reading). If you know your game will be a first-person shooter, your design choices have been heavily circumscribed.
When you playtest your game, the constraints are more specific. "Is this enjoyable" for my target audience? Is that too complex, or too simple? Does this element contribute to gameplay, or shall we "lose it?" Instead of that, what I'm talking about in this "origin" are additional constraints on the kind of game you want to make, imposed as part of the process of conception. I think that a good choice of constraints -- choosing more limited goals than "let's make a shooter" -- will lead to a better game.
Let me generalize. In many fields of art, periods of fairly clear "rules" for how to create art (for example, the sonata form in music or the three-movement form of a concerto) are followed by periods when there are no rules, until new rules are established. The greatest art comes from the periods when there are established rules. My example is primarily from music.
In the Rococo period following the Baroque, Johann Sebastian Bach's sons and others made good music, but not the great music we saw before and in the succeeding Classical era as defined by Hayden and Mozart. In the current era of "modern art" (painting) there are no rules. Not everyone thinks modern art is rubbish, but I think it won't get much respect in the future.
Yes, great artists often break some of the rules -- Beethoven comes to mind -- yet even they follow most of the "rules." When all the rules are swept aside, people grasp for new sets of rules (Schoenberg's 12-tone music?), but for a time the chaos results in little that is later recognized as great art.
Consequently, a designer usually benefits from additional limitations, whether imposed by a publisher or studio ("no foul language"), or by himself ("I want a one-hour trading game"). Even though a self-imposed limitation may ultimately be abandoned in the interests of making the game better, initially it focuses the designer's efforts and is likely to provide better results. There are always self-imposed limits, because you have your own preferences. And if you work for a game studio or publisher, you might find that you have to jettison some preferences: If they say "make such-and-such a game," you'll do it or you'll be out of a job.
In other words, self-imposed constraints are "rules" you use to try to help yourself create a better result, just as the "rules" for the various arts tend to yield better results.
In any case, be sure your idea origin isn't simply based on the game being "just like I'd like to play." You are not the audience. You are very unusual or you wouldn't be designing games. And the game you'd like to play has likely already been designed. My favorite game for 20 years was Dungeons & Dragons, but I have never tried to design a role-playing game. I like D&D -- why would I want to design something just like it?
It's too much time and effort to design a game just so you can play it. Game designers should design games that other people will enjoy playing. Most of the time, you'll like to play them, too.
Let me quote Sid Meier (Civilization, Sid Meier's Pirates) from GameInformer 182, June 2008:
"[T]here's a danger with some of the newer designers, a tendency to design the game you like to play. That game has already been designed -- we need new games. There's a loss of a little bit of that ‘sky's the limit, anything's possible' approach we had in the early days. We have these genres -- we have first-person shooters, we have real-time strategy. If you've played games all your life you've gotten these certain styles really beaten into you. To get people to think out of the box is a little harder these days."On the other hand, do not design a game you dislike to play yourself, at least not until you are very experienced. If you dislike it, why would you expect anyone else to like it? As you get more experience and understand players better, you may be able to design a game that appeals to a certain segment, even though it doesn't appeal to you. At some point this may be worth doing, to get you "out of a rut," to "think outside the box," but it's not something to be done lightly.