From that suggestion, John went into exploring the concept of a modular, tile-based character sheet. Each tile would have certain information about your character, but the way you arranged the tiles together would directly affect your abilities. For example, you might have skill trees or webs, and rearranging the tiles would let you connect skills together in different ways to get different bonuses or competencies. Or the game might use dice pools created by matching abilities to skills, so each tile would have particular abilities and skills written around the edges, and matching them up in different ways would open up different dice pools or change the size/number of dice available.
We then merged these ideas. The character sheet could go back to being a single sheet rather than multiple tiles, but it would still have some kind of connectors along the edges. The game map meanwhile would have other connectors, and you could change your abilities in the game by changing which edge of your sheet you connect to the map - or perhaps also by changing where on the map you connect to.
Finally we brought tiles back in, and abstracted the "map" out into a kind of game flowchart. The center tile of the map would represent the goal that everyone was working toward. Tiles radiating out from the center become the steps to the goal, and players connect their character tiles at the edges of the map. You might not fill in the whole map, and just have different tile paths, but no matter what, all the character tiles would have a path to the center. You also could have secondary goal tiles out along the edges or at the corners; each player could choose which if any of these to connect to as well. That way all the players would have a common goal to work toward but also separate personal goals that might bring them into conflict with each other. The choice of character tile sides to connect to the map/path/goal tiles would still directly affect your abilities in the game; you'd probably have some mechanism allowing the players to reorient their character tile, bringing different advantages and limitations into play.
John actually took notes and drew a couple diagrams during our conversation; I'm hoping he'll edit this post to add further comments if needed, and post the diagrams.
The very afternoon that John and I had this discussion, Joshua posted an idea called Web of Shadows, based upon his thoughts for a game system that would support the political and conspiracy gameplay that the World of Darkness line claims to be about but doesn't really support in its system. Joshua's concept included creating a literal web by connecting index cards with strands of yarn; the center of the web is the pivotal figure of the game, the "Chosen One," and through play the players' cards become connected to the Chosen One card and other cards representing various elements in the game. Joshua's concept ultimately is rather different from ours, but it's definitely very cool and worth reading both for its own sake and also to contrast with ours.