Superhero games are GREAT for one on one gaming. Think about it: Batman, Wolverine, Spiderman, Daredevil, Hulk, Superman, the list goes on, solo superheroes are always going on adventures by themselves. Work up a mini-campaign to be finished in one night, with the other guy playing the lone wolf superhero trying to finish the goal. It doesn't have to be all combat, there can definitely be character development, interactions, puzzle solving, etc. Plus the game can go any way you want if the superhero player decides to pull some wild and crazy antics! I used to do a lot of superhero one on one and they were always a great time. Some suggestions of rules systems are Villains & Vigilantes, Champions or Mutants & Masterminds.
Somewhat similar, the espionage genre is perfect for one on one gaming. One character can set up the mission and villains, the other can be the "James Bond" lone wolf or he can have a few characters "Mission Impossible" style to try and complete the task. You can go lots of ways here, and again this can have as much or as little interaction as possible. Game systems that are good here include the old Top Secret/SI system, James Bond RPG, and newer systems like GURPS or D20 Espionage.
Emulating the old Lovecraft/REHoward tales, a good horror RPG can be used for one on one action. In most of Lovecraft and Howard's horror stories, it was typically only one antiquarian who tried to find out what was going on in the old haunted mansion or shunned village. Call of Cthulhu works well for this genre. A good haunted mansion mystery (with the lone investigator, perhaps accompanied by an ill-fated bodyguard or two), spends the night in an isolated castle/farmhouse/island and the fun begins. I have done a lot of one on one horror and it's really great, because with only one player you can really set the mood that the guy is ALONE and NO ONE is coming to help him!!!
The great thing about any of these genres, you can have as few/many rules as you want. Heck, you can wing the entire thing if you want it to be pure storytelling! Maybe a few dice rolls thrown in for points where decisions need to be made, or when combat takes place, but otherwise it can be entirely just you and your buddy making it up as you go along within the framework of the superhero/espionage/horror genre...