Actually, it is a time for drinks before a hunt (like the fancy kind on horses with top hats), when everyone’s feet are already in their stirrups, hence the name! The traditional thing to drink was port or sherry. Remember how I wrote about how Scarlet is basically an etiquette manual? She goes to a hunt and participates in the stirrup cup in the book.
Apparently, even further back, anyone setting off on a journey on horseback would get a drink before departing.
It is also the name for the type of cup used for the event, often shaped like a fox or other animals. Commonly, the cup would end in a point because it was for drinking, not setting down! And servants would carry the cups around in a special tray with holes in it to pass them out to the guests.