lycopus, thank you. I'm not quite sure this is it. It looks very, very much like what I have, but the leaves are just a bit off. All the photos I see have leaves which are rather flat. The leaves on mine are more curled up at the edges, even with a slight ruffle to some of them. (Of course, that could be because it's in a vase of water on my table, and not outside in the sun in the garden! I'll take a look at the ones still in the garden next time I'm there.) Other than that, the images look very similar. I will leaves a few in the garden and see if they go to berry. That may help me identify it more firmly. It is indeed a nice looking plant. Perhaps I will try to transplant some out of the beds and along the woodland edge. Thank you! Dee

Looks like Pilea pumila.

It is a grape family* climber, what's left is figuring out which particular one. Spotting a likely seed source plant nearby, with its fully expressed foliage characters and other adult features would definitely help with this. *To which all previously suggested plants belong

You can still hand-pull that, but if it's spottily in other parts of the lawn as well, I might suggest spot-spraying with Weed B Gon Crabgrass Killer. Mix up the concentrate in a one-gallon sprayer (it's cheaper that way) and keep after it to avoid having the CG seed out.