What about "Guess Who?" game? http://en.wikipedia.org/wiki/Guess_Who%3F
I play it with my son and it seems like a perfect exercise in being specific. You need to be very observant and ask very precise questions. The competitive element increases its attractiveness. The set of cartoons/data to guess from is typically structured as lattice - there are many differently overlapping categories with subtly different frequencies - you quickly learn to exploit those to your advantage.
And of course you may experiment with game data. Replace cartoons with concepts. Or use artificially generated graphical shapes with different properties, e.g. number of vertexes, color, convex/concave etc. Or just spend few bucks and find/buy one of the replacement sheets on the web.