Black nightshade
(Solanum nigrum) is one of our most common introduced "weeds" here. Evidently there's a
complex of closely related species with edible ripe berries and leaves, many of which are cultivated. I took a group of cub scouts on a foraging walk last week and introduced them to the fruit and got responses from "meh" to "those are yum."
They can be confused with deadly nightshade, so be careful, and apparently even the safe species are capable of producing toxic alkaloids under certain conditions, probably related to soil types and stress. The Wikipedia entries go into more detail about this, so all I would advise is to limit how many you eat unless you know more about the strain you're picking.