Frank as in the late Frank Sinatra, the Chairman of the Board, an iconic American singer. Jack as in Jack Daniel's, the black label Old No. 7, an iconic American whiskey. As Sarah Feldberg explains in this article on the Tales of the Cocktail website, Sinatra's on stage endorsements beginning in 1955 caused Jack Daniel's to go from being a regional player to a global powerhouse.
If you want to emulate how Sinatra drank his Jack Daniel's, just remember 3,2,1 -- three rocks, two fingers of whiskey, one splash of water. In the mood for a cocktail? Go New Jersey with a Newark (Sinatra was born and raised in New Jersey), go New York with a Manhattan or a Brooklyn, or go with an eponymous drink.
With all of these ways to celebrate Frank Sinatra on his birthday (December 12) or any other day, what do you do? To paraphrase a line from one of his most famous songs, just drink it your way.