For a game to be about a certain idea or topic, it must focus on that idea. Focus can be created in a few ways. One is through detail: with charts, tables, die-types, modifiers, etcetera, you can cause more time and thought to be centered on one aspect of the game. (It is easy to unintentionally focus to much on the wrong things, by adding too much detail to them.)
But detail isn't enough for a truly central idea. Games are, after all, about decision-making as well. And focus is given to something very quickly by having to make decisions about it. Decisions must also have consequences, and the player must get feedback about them, to inform his next decision. Such feedback loops have to play out quickly enough so that many decisions about the topic are made in one game -- and even more evaluations about whether to change strategy are possible.
Randomness is mostly a separate issue, but too much randomness can smudge out the feedback cycle: consequences then become divorced from the decision, making it less important and less understood -- and less worthy of player focus. Similarly, the cycle of decision-making can be clouded by too many other, competing, cycles: a game can involve many types of decision, but only a few can be really central. Detail can also slow down the feedback cycle too much, by bogging the players down in execution (for complex board games in particular).
Failing to attend to the timescale of a feedback-cycle can create decisions that seem like afterthoughts, or which only really come into strategic focus after many repeat games. This is the criticism I have of initial placements in Settlers of Catan. The decision can have a large impact on the game (I would guess 25% to 33% of deciding who wins) but only happens once and is heavily clouded by randomness (in the form of turn order and the possible availability of highly desirable locations).
Poorly-fitting feedback cycles seem to be a greater danger in high-concept designs, which are inspired by a single abstract idea. To work, the idea has to be tied closely to the decisions players make: it can't merely arise organically from other mechanics, because the feedback cycle is then elongated (as the idea slowly coheres) and is clouded by an indrectness.
A game's subjects and its in-game elements should also match the scale of its central idea. Imagine a wargame is premised on the trade-offs involved in sacrificing troops for victory. The primary elements should probably not be individual soldiers, where randomness and details of performance are likely to dominate and swamp out strategic trade-offs.
Although a central idea must be well connected to other elements of the game, this can also go too far. If a decision in the central mechanic ramifies out to every other moving part, it becomes laborious for players to (a) come to a decision about how to use that main mechanic, and to (b) execute the decision on paper. Both of these things elongate the feedback cycle. But conversely, you don't want too many secondary elements which might bury the main mechanic. A designer must strike a balance between the number of different systems and their relative importance.