I've done a lot of one-on-one GMing. I lived with a family in 8th grade, and I had a 3-4 daya week one-on-one that went on for a year, as well as a few others early on, a few in college, and then 2 relationships and finally my wife.
System is the first question for a lot of reasons. The first is you have to find out what kind of game the player wants. One-on-one has some distinct advantages over multiplayer in that the player can REALLY get stuff done and can really focus on goals. Gritty or Epic power levels? Cinematic/narrativist or real?
Also, as per Vreeg's first rule of Setting design, ("Make sure the ruleset you are using matches the setting and game you want to play, because the setting and game WILL eventually match the system."), you need to know what game you want to run, and how much effort you want to put into it. Make sure the ruleset supports the type of play you are looking for. If you choose an encounter based game like 4e, don't expect the system to support heavy roleplaying or political gaming. Retroclones of 0D&D are better for games based on exploration, whereas some of the skillbased games are better when you want to add in more social interaction.
Talk to the player about lethality and setback ahead of time. These are some of the things that can end a game if there is a big disconnect between GM and Player. If the Player does not understand the stakes, they'll be upset if they lose an eye or die.
Discuss early on about how long this could last. Different games have different power growth curves. If the idea is to make this last a while, think about writing a little chapter guide for your self.
Decide the 'Sandbox' % early on. One of the tendencies that shows up in a one-on-one is the railroad. If you want to keep an open exploration, remind yourself of this often. A good sandbox game means that the Player will jump around and do different things; be prepared and patient here. Definitely have at least 2 major adventure ideas (at least) if you want to run something of a sandbox, so the player feels their choices matter from the outset.
Be ready for the player to collect a stable of NPCs!!! in a one-on-one, with a single Character, the GM's skill to create memorable, differentiated NPCs will be stretched. Get ready to act out mannerisms even more than than normal, and be ready for NPCs to be collected for a while.
A single player also means that the GM has to really take the time to create a stable yet 'World-in-Motion' style home base, especially in the earlier games. Take the time to write good notes ahead about the home town and every NPC therein that they meet. Know where mercenaries are found, how informtion is shared (town crier? Broadsheet? Bardic Network?), and where different supplies are found. Include a few places in the beginning that the player cannot afford, so that down the road the feeling of accomplishment is that much sweeter.
OK, back to work. I hope that was useful!