Lady Caroline Stanhopes birthday
There were two people known as Lady Caroline Stanhope in Thompson's time - mother and daughter. The mother was originally Caroline Fitzroy and her first child on her marriage to William Stanhope, 2nd Earl of Harrington was also named Caroline. The daughter died in in 1767 aged 19. The birthday of the dance title could have been for either of these characters. In time, more detailed study might reveal which one.
More information.
Lady Caroline Stanhope the elder