Passive solar power is free and plentiful. "Passive" means you are simply letting sunlight into your outhouse, usually through a window. Advanced methods of passive solar power will collect and even amplify sunlight as it enters the outhouse. Because you want privacy in an outhouse, windows should be made of distorted or frosted glass. Certain roof structures, such as tubular skylights and roof monitors, can amplify light but cut down on the sun's heat, an important feature in an outhouse that will be more odorous as the temperature rises. Passive solar power will work all day, every day --- unless your outhouse is in a particularly shady or darkened area.
A small solar panel, if it has clear access to sunlight, can easily provide enough energy to power one light bulb for an outhouse. A 5-watt solar panel, for example, connected to a 12-volt battery can light a 5-watt fluorescent bulb. A 5-watt fluorescent bulb puts out as much light as a 25- to 30-watt incandescent bulb. Add a lampshade for a decorative, practical, self-contained solar light system. The cost of a 5-watt solar panel begins around $25, and these systems can be added to any existing structure.
Another option is a portable solar lantern, kept outside the door of the outhouse by day and brought inside only during use. These lights can multitask, providing light during camping trips or nighttime hikes as well. Some of them last longer than others, and they will function best if turned off when not in use. These range in price from about $25 to $75. You also can get small garden solar lamps for lighting the way to the outhouse.
There are alternatives to solar power for lighting your outhouse. For example, you can hook up an exercise bike to a battery to generate electricity. Advanced light tubes can be used to track the moon and harness its light. You can build an outhouse that is easy to feel your way around, eliminating the need for artificial light. You can also just use your night vision: after half an hour in the dark, your eyes have adjusted and you can often find your way around with minimum light.