Old enemy identified
J.D., one of the botanists from my PR days with the Gulf Coast Research Laboratory, suggested that the photos I emailed him looked like Phyllanthus urinaria, common name chamber bitter. Thanks, J.D.
Other common names are gripeweed, shatterstone, stonebreaker and leafflower, according to Wikipedia. Here are parts of Wikipedia's description that match what I have observed in our own garden:
The plant, reaching around 2 feet, has small alternate leaves resembling those of the mimosa tree. When touched, the leaves fold in automatically.
A photo that accompanies the Wikipedia entry here boasts reddish fruit that I have not seen before, but then this is the first season I have ever seen the plants get this big and so extensive. I guess my post-stroke mulching discouraged them in the summers of 2011 and 2012.
The characteristic of how the leaves fold up when touched is probably why I retain a vivid memory of my mother's showing me the plant when I was little. I still cannot remember exactly what she called it, but it was something like "sleepy weed."
Of course, when I googled "sleepy weed," only citations related to marijuana came up. I don't want that weed in the family vegetable patch, either!