Regardless of your opinion of university uprisings and Haight-Ashbury, the 1960s were revolutionary. Not least because a game called Stratego was first published in the United States, and while it was based on much older Chinese and European games, it introduced Americans to a concept they hadn't yet embraced—that play needn't be mindless.
Since then, strategy games have evolved into far more complex and sophisticated creatures than their Stratego ancestor, though they've retained the basic elements. They're generally built around a theme (Stratego centered around defusing bombs with engineers; modern games let players pretend to be world conquerors, space truckers, or cavemen), logic and planning are central elements, and play is moderately paced.
Agricola is a favorite around here. Players are Roman farmers with a little land and a little lady. The goal is to move from mere subsistence to prosperity through building fences, raising livestock, growing crops, and producing progeny. The Ticket to Ride family of games lets you build your own railroad empire by buying routes and lines and generally glorying in capitalist exploitation.
For more bloodthirsty gamers, Merchants and Marauders lets you choose between privateering and peaceful merchant shipping in the Europe-dominated 17th-century South Seas. (Making all players speak Piratese during gameplay is an interesting variant.) Rattus puts you in the middle of the Black Death as rats spread through your territories and those of your enemies. To win, be the player with the most people still alive by the end of the game.
There's usually some element of chance even in the most complex strategy game, though play always centers around choices players make and doesn't hinge on drawing cards, rolling the bones, etc. Strategy games might not be the best choice for kiddos, but if playing Sorry sounds about as fun as walking barefoot over a floor strewn with Monopoly pieces, give one or two of these titles a shot.