If I want to add light to my chicken coop during the winter, how much do I need to add?
You needn't use a very bright light; studies have shown that laying increases with about as much light as a candle. Even something like a 4 watt nightlight would likely work.
It is usually recommended to only add light in the morning, though, to avoid stressing them at night. The reason adding light in the evening can stress them is that in normal circumstances (with natural light), the light fades gradually as the sun sets, and that gives them time to find their places on the roost and prepare for sleep. By contrast, when an artificial light goes out, it is very sudden, and they don't have the opportunity to wind down and find their accustomed places on the roost. They may be on the floor of the coop eating or drinking when the lights go out, and unable to clearly see how to get back up to the roost, knocking into one another or having trouble jumping up in the dark. If you must add light, do so only during the morning hours, and allow the light to naturally fade in the evening as the sun goes down.
Do consider allowing your birds to have their natural break in laying during the winter, too. The problem with keeping the daylight constant is that it may cause your birds to molt late, in the dead of winter, when it is cold and they need their feathers the most. Chickens will molt annually, regardless of the light situation; however, it is normally the change in daylight that triggers it (not temperature changes). If your chickens don't have that trigger from fading hours of light, they may hold on for several months before finally molting at a time when it is really too cold to be without feathers.
The bottom line is that if you must add light at all during the winter, you want to wait to add it until after your birds have had their annual molt, and then you will want to add it at the beginning of the day only.