My wife and I have gotten a lot of mileage out of playing these 1v1 games:

  - Twilight Struggle  
  - Hive  
  - Patchwork

Choosing one really depends on mood. Twilight Struggle is the kind of game you have to buckle up and commit to playing for a while, but it's worth it. Hive and Patchwork are more bite-sized - both have enough complexity to be interesting, but with a much lower time commitment. You also have to refer to the rulebook less often with those two games. :)