John, I don't know about particular plants that would grow in the shade, but plan for lots and lots of leaf mulch in your rain garden. I inherited a small, lined pond that the prior owners located under a bunch of big leaf maples. It is a leaf bog. Now, because it is lined and the liner is piles with heavy rocks, I haven't gotten around to fixing it. But not a whole lot seems to be growing around the edge in this area due to the heavy layer of leaf mulch every fall. But, you know who LOVES it? The local frogs. I have so many frogs in my garden this year. Just design with an annual heavy influx of mulch in mind.
If you don't need them to be edible, take a look at native plants for your area. Native grasses do great in rain gardens.
My goal? To create a wildlife habitat on our surburban (almost rural) property using mostly California native plants and inspire others to do the same.
Also to learn how to grow some edibles - more than the herbs I have grown before.
I'd assume that other cane plants (like raspberries and domesticated blackberries would do well).
Lingonberries like it wet, and might tolerate the shade. Cattails are another edible that will probably do okay, even in the shade. Bunchberry and violets and miners lettuce should do well on damp ground, and they also like the shade--all three are edible. Devilsclub is a medicinal, horribly pokey large shrub that likes the shade and wet, and might try to take over.
I know there's more, like watercress and waterchestnut grow in water, but I can't remember how they do with shade.