Choose something appropriate for your area that you find desirable. Note these are edibles for the most part. There should be plenty of non-edibles tolerant or even thriving with only partial shade. Of course some things may go dormant for winter...or even summer. I think I would try the alpine, musk and lipstick strawberries. Or plant a pecan tree to get the area overhead shade (dappled) in the summer that I could then plant under..

SHADE TOLERANT

Partial Shade
May have less fruit.
Alpine Strawberry
Bamboo (some types)
Dogwood
Elderberry
Evergreen &
Red Huckleberry
Filbert
Gooseberry
Hydrangea
Jasmine
Lipstick Strawberry
Lingonberry
Magnolia Vines
Musk Strawberry
Paw Paw
Pie Cherry
Salmonberry
Serviceberry
Sugar Maple
Thimbleberry
Full Shade

Bamboo
(in sunny climates)
Currants
Carolina Allspice
Evergreen Huck.
Flowering Currant
Wasabi
Highbush Cranberry

Here is a link that might be useful: List from Raintree Nursery