I figured the best way to repair the rain barrel was to drill a hole where the nozzle had been and insert a new nozzle. I found something like this at Home Depot.
This is a barb on one end, suitable for slipping a hose over it, and 3/4 inch threads on the other end. So now I needed to drill and thread a 3/4 inch hole. Turns out I have a tool to create the 3/4 inch threads from my work with the PVC underground irrigation work:
This has 2 or 4 different threadings, one of which is just what I needed. So after drilling a smaller hole first, and then expanding it with the threading tool, I was able to get the adapter screwed into the rain barrel, and re-attach the hose.
An alternative, if we need one in the future, is to put in a brass adapter, but with a plastic rain barrel, a plastic adapter seems reasonable.