"hair of the dog that bit me" comes from the old belief that if you were bit by a rabid dog, rubbing the fur on the wound would prevent getting ill. The dog is alcohol, and the hangover is his bite, the hair of the dog is the 'little bit" you consume to feel better. Since they didn't know you were merely extending being drunk, they thought it cured a hangover.
Hair of the dog that bit you. The saying originates from the old belief in the treatment of a Rabid dog bite. It was believed that if you put hair from the dog that bit you into the bite wound, it would prevent 'evil' consequences.
Ask a dog. Quite a few of them here on EP I suppose.