I live in San Francisco, and this plant has invaded my lawn with a vengeance. Pics are below. Its leaves/stems are slightly fuzzy, and its roots are kind of woody. It pops up all over the place and I have a sinking suspicion that they are all connected via a common root structure.
You'll probably be able to tell when the flowers come out. I also live in SF in the Mission, and the chickweed always appears in my garden in the springtime for a couple of months, but dries out and disappears after that. It does get much bigger (8-10") in my yard, the stems have an elastic feel to them, and I put it in my salads, it's really good for you and tastes great. My chickweed is a lot more fragile-looking. I can't tell for sure from the photo if it's definitely chickweed, but the little white flowers will give you a better idea. The other plant that looks similar and also grows around these parts is Scarlet Pimpernel. Here is a blog post with a pic of my SF spring chickweed: http://www.lazycomposter.com/spring-greens. There are some flowers in the pic, but you can barely see them, they're so tiny. That root in your picture looks awfully woody for chickweed.