I have never run specifically a single-player campaign - but once in a while I play one-on-one with my players, usually when there is a subplot involving only one player, or when the others are not available.

I keep specific session ideas for the one-on-one plays, since as goto124 stated, you can go deeper in the roleplay when playing just with one character. You can focus on the PC-NPC relations, PC character, his backstory, subplots and also - the player has time and space to really deep into the character.

For one-on-one campaing, I would suggest a system, that provides opportunities for single players - classless systems work better due to possibility of creating a character that can operate without a group or with a single or two NPC helpers; systems which give you fully functional character (i.e. not starting at level 1/lower on the competence ladder) are also advantagious. Focusing on the roleplay aspect is good too.

From my experience, Shadowrun works fine as one-on-one, since you can get NPC runners to assist you easily, while still keeping focus only on your character, and provides lots of possible roleplaying experiences.