Portulaca grandiflora Moss-rose

A very lovely plant, but a summer annual (dies off in winter).

You might look at:

Lippia
Dymondia margaretae
ground-cover Thymes
low growing Arctostaphylos
perhaps a short sedge such as Carex pansa

or a mixture of all or some of these.

I think that if you plan on doing this, you may want to consider stepping stones to give the area a break. I suggest planning a trip to the race track in Berkeley and picking up manure and spreading it on the area in fall, let the (hopefully) rains wash it in and plant in the spring. Go to your nursery and talk to them or while you are at the race track, take a visit to Berkeley Hort and talk with them. Why not creat areas and vary the plantings, put in some salvia, cat mint, geraniums and thymes. It could be quite beautiful, use a low amount of water and still satisfy your eye. There are even ground cover roses, once established that do not need much water.
There is much you can do.

Definitely improve the soil before you put anything in. Without doing so, you will just waste water and fertilizer, not to mention the cost of plants.

Bluesibe's ideas for making a cohesive plan first are great ideas. Be aware this will require more gardening time than the usual 'mow and blow' care of a lawn, but it's very rewarding and a lot more fun than an exercise bike, LOL.

Do NOT steal from a city park. This is no time to add a felony charge to your employment record. If you live in SF I can't imagine you have a large area to cover, and groundcover plants are cheap in flats at OSH or HDepot.

Lawn daisies are a weed in the grass of lawns. Please read about them. There are all over the place not just in golden gate park, and if I was to remove some of them, I am only weeding. There were not planted by the parks. But, the ones you buy are not hardly types. I think they are over looked at a ground cover. I have larger plants are roses around the edges, but the center part needs something that will not wrap around the garden hose and will keep the dirt down. I am not sure they would make a great ground cover. I don't think think they grow close enough together. But, they look nice. I don't know if it really viable as a ground cover.

Here is a link that might be useful: Lawn Daisies

Dymondia margaretae looks like a good choice. It's in the composite family, but I wonder if it would be hard to find in location nurseries. I have to make a few phone calls. I can't start until next year because I have to prepare the soil. The lawn was so old all the soil has gone very poor.

Here is a link that might be useful: Dymondia margaretae

I'm not saying the law is going to hunt you down where you live or harass you; it's likely the park gardeners won't be around to notice since there's so few staff left after severe budget cuts. But although you might call it weeding, in the eyes of the law, it's theft, like it or not.

There are many plants that would do well assuming your dog won't dig them up? - some dogs are manic for digging and others aren't, so I'll assume yours doesn't and the plants can establish themselves well through the winter. We replaced our soil with top quality compost and mulch heavily. I live in the Oakland hills so a bit warmer than you but still at the edge of the fog. We water every two to three weeks, depending upon how many 'heat spells' the Bay Area gets during the summer.

The very toughest plant I have is polygonum. A true groundcover, it's actually invasive so only suitable if well-corralled by very long stretches of concrete. And I mean long, I've had this pop up up to 10' away although no further than that! With good soil and mulching it can take a surprisingly small amount of water. Here it is mixed with Oxalis siliquosa. Polygonum is the darker variegated arrowhead-shaped leaf with tiny pink globular flowers:
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There's nothing more drought-resistant than Stacchys byzantina, or lamb's ears. "Helen von stein" is a variety without flowers, but lacks the silvery hue this one shows. The accompanying plant is Tanacetum, a lovely plant but harder to find:
{{gwi:558194}}

I don't especially like Vinca major or minor. They take quite a while to establish and need a fair amount of water. The two variegated forms are very handsome, however, and when the long stems (think ivy-like) throw out their flowers, they can be a beautiful addition:
{{gwi:558196}}

Lotus berthelotti comes in yellow, orange or red forms, and has beautiful foliage. It does demand more water and feeding, however, to bloom its best:
{{gwi:558198}}

These are commonly called ivy geraniums, a complete misnomer since they are actually pelargoniums. This is a trailing groundcover form, but I have only seen it in this color, oddly enough:
{{gwi:558200}}

If you have room to break up the groundcover look with some subshrubs, osteospermums are truly amazing plants. They are drought resistant, flower almost continually, reseed with abandon but are easily pulled out if unwanted. They come in a number of colors now and generally stay 1-2' high:
{{gwi:558202}}

Like L. berthelotti, the following have slightly higher water requirements, but are still manageable and would like your SF temps.

Lamium maculatum 'Anne Greenaway' (the little white daisy-like plant in the top LH is the Santa Barbara daisy, also a contender for tough, drought-resistant; but more of a mounding plant than a groundcover):
{{gwi:558205}}

Lamium maculatum 'Pink Dragon' (there's a white flowering form as well):
{{gwi:558207}}

Oxalis siliquosa 'Copper Sunset' and a variegated yellow plectranthus, forget the variety name but has pretty little spikes of pale lilac flowers:
{{gwi:558209}}

Thank you for the beautiful photos. I had some Polygonum in my front planter as a ground cover and I have been trying to remove it for six months, but each time a million tiny sprouts grow back. I don't know if I will ever get rid of it. I have to keep retilling the soil. I really like the Oxalis Sunset. I bought several of those but they never took, like the Iron Cross Oxalis. I have actually started several others from just the one and it's really hardly, but I don't how it will handle traffic. The sunset oxalis is so much more beautiful. I got it at east bay nursery, but I think it's more succulent and will crunch up if walked on. However, all my sunset oxalis died. I had lambs ears also, but it grew kind of tall. I remember it was hard to get rid of. I got it at a swap, and then I wanted to get rid of it, and it keep coming back. I really like that Lotus berthelotii. I may end of using a number of different ground covers for variety. Thanks for all of those suggestions. I hope the dogs won't dig it up, and they probably won't. I think lawn daisies could be a ground cover. I can find them in non park areas such as there is some near Glen Canyon. I don't know why people don't grow them as ornamentals. The ones they sell are bigger flowered and not very hardy, but it might not work because it may not knit together enough to keep weed out. I think they look cheerful in the lawns, but they really are a weed type like a dandelion, so I never tired to add them yet.

I don't want a grass type because I have so much pod annual that it's too hard to weed out if I have a grass type cover. I would totally not suggest anyone do a mass planting of the Polygonum. It makes millions of seeds and each seeds stays viable for a very long time. So, if one ever wanted to sell the house and put a lawn back in it would not be possible.

Here is a link that might be useful: Iron Cross Oxalis