Day lilies are eminently edible! The flowers and flower buds taste like a sweet, flower-y version of leeks, if that makes sense. Really tasty stir-fried. Other parts of the plant (tubers and young shoots in spring) are also supposed to be edible, but I don't have first-hand experience there.
One small note of caution, though: I seem to remember reading in The Forager's Harvest by Samuel Thayer that some (but not all) clones of Hemerocallis fulva specifically can induce nausea. I think he said that for clones one hasn't tried eating from before, one should exercise caution. By contrast, all clones of yellow daylilies (H. lilioasphodelus) are supposed to be safe.
Edit to say that I've personally only eaten yellow daylily (H. lilioasphodelus) because of the above. Also, I was poking around online just now and found
this article, which suggests that eating the roots of at least some of the species might be a bad idea.
Severe toxicity of Hemerocallis has occasionally been reported after ingestion of root material (Xiao et al. 2010). This is probably due to the presence of stypandrol (hemerocallin) in root tissues, a neurotoxic compound also found in the related genera Stypandra and Dianella (Wang et al. 1989).