The story
From the act of tying or even nailing a pigeon to a stool to act as a decoy by hunters. Alternatively, from an alteration of stale (“decoy”) + pigeon.
Etymology from Wiktionary, CC BY-SA 4.0.
stool pigeon
Today it means: A pigeon tied to a stool by a piece of twine, used as a decoy to entice free-flying birds to the ground to be shot or trapped. Full definition →