Ask Mr. Smarty Plants

Well, guess what? Hydrocotyle bonariensis (largeleaf pennywort), which is probably the one you have, seems like a nice, natural lawn replacement where there is too much shade and moisture. In fact, as we looked through references to this plant, it seemed that many, if not most, of the sites knocking dollarweed were sponsored by herbicide manufacturers. Go to this site emilycompost and get some positive feedback. If you call it "pennywort", which is another common name, you'll find there is a lot less negative comment about it. Another native plant, Dichondra recurvata (oakwoods ponysfoot), looks very similar but is not related. Dichondra is widely used, especially in California, as a lawn plant. It's very ironic that you are probably replacing one of two invasive, non-native grasses-St. Augustine and Bermuda-with a native, attractive, low maintenance ground cover.

In terms of propagation, since Hydrocotyle bonariensis (largeleaf pennywort) is a native plant which volunteered to live in your lawn, I think you might just let Nature take its course. It is a perennial, with stems creeping and rooting at the nodes; it spreads by rhizomes. It's going to do better in the shady environment and around your flagstones than a competing grass. It does bloom, from April to September, and probably will seed out from that. If you can convince your husband that it reduces his mowing, it's evergreen, needs less water and is going to be there anyway, you probably will have no further problem. If it chooses to get into flower beds, you'll probably want to remove it, but it's certainly less invasive in that respect than the St. Augustine and Bermuda.

Dichondra recurvata

Hydrocotyle bonariensis