Try elfin thyme. It should be hardy down to zone 3, grows about a centimeter tall in a dense mat, and is drought tolerant. Likes full sun, is hardy enough for some foot traffic without dying, and it makes pretty little purple flowers in the summer.

You could go with clover tho u will have bees. It stays green in hot dry weather. Just a thought.

I got rid of my lawn a few years ago & tested lots of available groundcoveres. The two that have survived are the sedum acre -- the hardy yellow one, & my favorite the creeping potentilla -- it has cute little yellow flowers & has surved several Iowa winters so far. My other groundcovers just didn't have the stamina to survive over time.

Good luck
Susan

Perhaps, you asked the question elsewhere & got a reply for how to get rid of the grass. If not, I suggest you layer over the turn after mowing short. Use cardboard, 10 thickness newspaper, or office paper 10 sheets thick overlapping and top with mulch or compost several inches up. Then plant in that. Many groundcovers are fast growers, but you will need to keep mulch and patrol for weeds until they fill out.

Stepables advertises many groundcovers as lawn substitutes. YOu might want to try out a few before you make a large purchase in case one doesn't grow well for you or you prefer the growing habit of another.

I don't like all the groundcovers and tended pots of them are different than run loose in the ground.

There should be a substitute that you'd like and you may want to install some nonplant groundcover such as rock or patio as well. Depending on how you want to use that space...

We installed a small green house & sitting patio out back as part of our lawn and then expanded with paths of wood chips and additional woodland gardens. The mowing area has significantly been reduced as well as the enjoyment of the backyard increased. I enjoy a nice patch of grass, but rarely sit out on it. I do however sit at the bistro table or the bench during nice weather. I enjoy the view of the patio daily rain or shine.

I would remove more grass in a heartbeat if it is another project like we did for our backyard.

Some cover that does not grow is most permanent.

There is a natural process called succession, that is as difficult to fight as the force of gravity.
The fight to keep a lawn typifies succession, and requires constant work, money, chemicals and mowing, just to keep the natural vegetation from moving to a climax forest.
Mulch renewal can be a once a year project.
Landscaping blocks or concrete might hold succession back for years, but even then, leaves can accumulate, rot, and succession will start.