I wanted to see what other people had to say before commenting in this thread, on the grounds that it avoids typig stuff people will mention anyway!
I may be unusual as a DM/player in actually preferring the one-on-one game to group play. (I tend to prefer use of the term 'one-on-one' to 'solo' play, since it makes clear that we don't mean involving only a single person, like the old adventure game books.) It's probably because I started out D&D in a small village, where there
were only the two of us playing, so we had to take turns as DM. But it's also because I really enjoy role-playing, and when there's only one other player you can never have to worry about boring others when you are deeply involved in role-play with another person. As such, many of my best role-playing moments have come from one-on-one play. I recall one poignant moment, where an NPC friend of the PC died, where both of us were choking back tears!
So how to run one-on-one play? Many points have already been made about choosing the right storyline, background and style. One I will comment on is that I find playing multiple PCs to be fairly unsatisfying - both of the characters come out 'flatter' and weaker because of it. It's not impossible, but it loses something of the fun for me. Instead, give the PC an NPC companion, someone they can actually like and learn to regard as a friend if possible. That way you get something of a party dynamic, as well as adding to the capabilties. You must be careful as DM not to make that companion domineering, but actually that isn't hard to do. Treat the NPC as a person, rather than as a source of power, and they take on a life of their own.
It's harder to do a dungeon bash with one player, but I assure you it's possible. You have to be very careful what you throw at the PC, of course, even where there is an NPC companion. It's probably best to use small dungeons, with clear goals, rather than anything huge.
Additionally it's quite hard to use standard adventures as written, even if you use ones that are for parties 2 - 3 levels lower. But I find you can use them for maps and ideas, whilst reworking things for a single PC.
More political games
do work very well for a single PC, if that is your and your player's taste. Even if you don't want to do an exclusively political style campaign, you may find yourself including more personal interaction with people anyway. Whereas in a normal game you'd gloss over that innkeeper, you might choose to play him out a little, flesh things out a bit, give yourself more small adventure hooks.
If you have any speicifc questions do bring them back to the board - I am sure there are plenty of us who'd be happy to help. And do give one-on-one play a try - it has a somewhat different feel to any other RP style, and can be very satisfying.
Good luck!
