I've had good luck with strawberries in the past. They grew really well in my backyard under a large oak (I was living in Ohio at the time). We also had some hot pepper plants and tomatoes planted in the same shady spot and they did just as well =) Plus, made for a tasty treat later! =D

Some of the mints like shade.

You could try lettuce - it will be leggy but it grows.

leggy?

Leggy ... tall and skinny leaves. Tastes fine.