A birthday suit was, originally, a suit of clothes worn on the king's birthday. The term first pops up in Swift's Modest Proposal (1729), where the approach of the Day of Judgement encourages some rich ladies to cancel their orders at the dress-makers:
Three of the maids of honour sent to countermand their birth-day clothes; two of them burnt all their collections of novels and romances, and sent to a bookseller's in Pall Mall to buy each of them a Bible, and Taylor's "Holy Living and Dying." But I must do all of them the justice to acknowledge, that they showed a very decent behaviour in the drawing-room, and restrained themselves from those innocent freedoms, and little levities, so commonly incident to our ladies of their profession. So many birth-day suits were countermanded the next day, that most of the tailors and mantua-makers discharged all their journeymen and women.
A mantua, by the way, is a kind of loose gown. Anyway, the phrase birthday suit has such an obvious suggestion of being as naked as the day that you were born, that Grose's Dictionary of the Vulgar Tongue (1811) has this entry:
Balum Rancum. A hop or dance where the women are all prostitutes. N.B. The company dance in their birthday suits.
I am now off to dance a balum rancum.
And anyone who works out what the pictures are is terribly clever.