Many of my daylilies are grown in part shade. At some point, however, it gets too shady for most daylilies to put out more than a few blooms, if that. At least, that has been my experience here. (A big surprise to me this season was that 'Let Me Be Clear' sent up rebloom scapes. It is growing in a one gallon pot in an area which gets maybe one hour of sunlight a day, during the peak sunlight hours of the year. It is a winter-bred dormant diploid, though, so it may not do well in southern gardens.)
A few daylilies here which are grown in part shade, which seem to perform well (or at least well enough):
Black Falcon Ritual (or (long story) possibly a seedling thereof)
Insider Trading
Madge Cayse (gets a fair amount of shade so does not rebloom here)
One Fine Day (blooms E-EM, then again in late summer/early fall)
Osterized
Sears Tower
It has been my observation that some daylilies which reliably rebloomed at our old house (which had little shade) do not do so here. Two that come to mind are 'Bamboo Spitfire' and 'Green Dolphin Street'. My takeaway from that is that while you may get a decent amount of initial bloom in part shade, you may - with some cultivars - lose rebloom.
I've read that many of Bill Munson's cultivars were shade tolerant. The only one of his that I still have is 'Burmese Buddha'. While it blooms and some years reblooms, I can't say that it is putting on that great of a show. I'm not sure about bud count changes (it was in a little shade at the old house), but the % of polymerous blooms that I see here is much lower, almost non-existent. To be honest, though, that could be a result of a change in other growth parameters. At the last house it was in a whiskey barrel (containing whatever potting soil and compost) and got hand watered with a hose at least 3x a week. Here it is in the native clay soil, and besides getting a lot of shade, is only getting irrigated twice a week.