Two native plants that volunteer in lawns in many parts of North America, both treated as weeds by many homeowners, in fact make excellent groundcovers. Both will take quite a bit of foot traffic; both are naturally fairly tall (2-3 feet) but can be mowed to whatever height you desire. Both will grow almost anywhere as long as they get sun and spread readily, outcompeting most weeds. What's more, they're usually available for free.
One is common yarrow, Achillea millefolium. The feathery foliage is reason enough to plant it, but yarrow is also aromatic. The flowers, which attract butterflies, are flat-topped and whitish, though they may have a pinkish or purplish tinge to them. Yarrow will often bloom even if mowed several times during a season. Though the yarrow in my lawn had only just started to grow, I recognized it by the tiny green fronds, which almost look like ferns, attached to stout rhizomes.
The other is aster, Aster spp. There are so many wild asters that I can't be sure which one this is. Chances are, however, that it's Aster novae-angliae, New England aster. Almost all the asters work equally well as replacements for lawn. Like yarrow, they will often bloom even when mowed, and the white-to-purple flowers attract butterflies. This plant hasn't even started to show foliage yet, but I was able to recognize it by the thick rhizomes that have a purplish tinge to them.
So there I was, on a chilly day in early April, carefully patting the roots of "weeds" back into the soil after discarding all the lawn grasses I could find. I kept at it until the sun started to go down and the wind picked up. When you can no longer feel your fingers, it's time to go inside.
What do I have against lawn grasses? That's a topic for another day.