The usual groundcover thyme is creeping thyme. It works well at suppressing crabgrass (a fairly easy task), less well at wood sorrel (a much harder task). It will spread into the lawn, and generally seed around. It may be aggressive enough to slow the spread of the daylilies, though it isn't tall enough to interfere with established plants. Personally, I'd pick veronica Georgia Blue over thyme for this purpose. It's more attractive in flower, and a better level of aggressiveness. Chrysogonum also would be nice, geranium dalmaticum is a very short perennial geranium that works as a groundcover.

In general, I'd say you want a shade tolerant groundcover because you want something that will grow under the daylily foliage, and not compete well with lawn grass.

I use creeping thyme grown from seed in a couple places in my gardens. It is agressive but easily pulled. This year with our good rains it has spread far and wide. I pulled a bunch out and cut it back before blooms could set seed. I like the low green look of its new foliage best. Took about two weeks to look nice again.

In another place I have wooly thyme which has spread more than I expected and has stayed nice and low. I am considering replacing the creeping thyme with more wooly thyme.

I have read that weeds can still grow in thyme and have pulled a couple of maple tree seedlings and other tiny seedlings out of mine. You could put leaf mulch around the thyme and as it grows longer it will cover the leaf mulch (which helps prevent weeds.)

Forgot to mention that I use hardy geraniums, 'White Nancy' and other varieties of lamium as well as a creeping sedum for ground cover.

Oy, when I inherited the bed there were hardy geraniums planted in it, and it's been the absolute devil to eliminate them. Not that I have a big problem with Cranebills, but the person who put these in was both color-blind and had no sense of where to plant things. I agree - they ARE effective at covering a big patch of dirt!

The White Nancy lamium looks like a good bet to plant in the part of the bed with the clematis - it's Rebecca, and blooms a deep, deep red.

And the Veronica Georgia Blue looks AMAZING! I wasn't familiar with that one at all. I had Veronica Goodness Grows in a bed with May Night Salvias at my old house - I didn't know that there were groundcovering, low-growing Veronicas.

I did see Wooly Thyme at the Maine gardens - that was a first for me, and it looked good (and felt even better). I don't mind pulling the occasional weed, but this summer - even with our record heat and a big drought - I'd come back from a week vacation and find the entire bed utterly overrun with weeds. If I can get this knocked back to an hour a week maintenance (without engaging in the Mulch Wars) I'm going to be really happy. I know that eventually, the daylilies will fill in, and I'll only have to concern myself with weeds that make it to 2 feet tall. :)

Thanks for the great suggestions!

Sedum John Creech makes a very handsome ground cover and can tolerate considerable shade. The best thymes that I've found--and I've tried 8 or 10 different ones are the very flat ones from High Country Gardens. Occasionally I do have to pull them back from around perennials. I've never seen a single seedling.

Veronica Georgia Blue. I inherited this plant when we bought our home 10 years ago. This, along with the old Ditch Lilies, were the only flowering plants on a 40 year old property - 2/3 acre.

I have managed to remove it from the gardens which I've created and it has nestled quite nicely in the lawn where today it covers an area 10 feet in diameter. In a Daylily garden? Sure! It would do well.