Over on A Paladin in Citadel, Aaron was musing on a possible fatigue rule that might be tied to hit points somehow. Having recently put down some of my own thoughts on hit points, I wanted to chime back in briefly.
I'm a believer that simpler is often better. Not always better, and not always more fun, but keeping things simple seems to work well most of the time.
With hit points, I feel that they already represent fatigue since it can be presumed that the character is doing their utmost to stay alive in combat no matter how pathetic they presume the enemy with the pointed stick might be. When those points are down it represents them being somewhat fatigued by the effort. Thus for me I don't worry about an additional fatigue mechanic.
Labyrinth Lord and other B/X versions already include an additional fatigue mechanic when a group has been exploring for an hour without rest.
If you were going to include a hit point related mechanic I would recommend the simple approach and give a -1 to everything at half hit points and below.