When someone says a game is "elegant", what do they mean? I'm not sure, so I've done a bit of investigating.
Is it used much? In my Info Select database, which includes my own notes about game design and teaching, and material that I've scraped off the Internet about those same topics in the past seven years, there are 84 notes containing the word "elegant" and another 34 containing "elegance". Clearly the term is used a lot in conversations and writing.
What about dictionary definitions of the word?
1. tastefully fine or luxurious in dress, style, design, etc.: elegant furnishings.
2. gracefully refined and dignified, as in tastes, habits, or literary style: an elegant young gentleman; an elegant prosodist.
3. graceful in form or movement: an elegant wave of the hand. [my emphasis]
4. appropriate to refined taste: a man devoted to elegant pursuits.
5. excellent; fine; superior: an absolutely elegant wine.
Synonyms: 1. See fine. 2. polished, courtly. [my emphasis]
World English Dictionary
elegant — adj
1. tasteful in dress, style, or design
2. dignified and graceful in appearance, behaviour, etc
3. cleverly simple; ingenious: an elegant solution to a problem [my emphasis]
Elegance is a synonym for beauty that has come to acquire the additional connotations of unusual effectiveness and simplicity. It is frequently used as a standard of tastefulness particularly in the areas of visual design, decoration, the sciences, and the esthetics of mathematics. Elegant things exhibit refined grace and dignified propriety. [my emphasis]
So could we say, for games: "A solution to a design problem that is seen as ingenious or cleverly simple, polished, and effective?"
At some point I wondered what the difference is between "elegant" and "clever"? For me, something can be clever without being worth doing; something that is elegant is likely worth doing. So I might see a game and say "that's a clever juxtaposition of mechanics", and still not think the game was worth bothering with. I would tend to think of games that model something in interesting or intriguing ways as elegant, whereas games that don't model something may only be clever.
So one man's clever may be another man's elegant.
adjective, -er, -est.
1. mentally bright; having sharp or quick intelligence; able.
2. superficially skillful, witty, or original in character or construction; facile: It was an amusing, clever play, but of no lasting value.
3. showing inventiveness or originality; ingenious: His clever device was the first to solve the problem.
4. adroit with the hands or body; dexterous or nimble.
ingenious, talented, quick-witted; smart, gifted; apt, expert.
There is no Wikipedia entry for the word "clever".
A last expression of the idea of elegance, from the point of view of design:
"A designer knows he has achieved perfection not when there is nothing left to add, but when there is nothing left to take away." --Antoine de Saint-Exup'ery
When you achieve this "perfection", you also achieve elegance.
So what do you mean when (if) you describe a game, or part of a game, as "elegant"?