The actual name for the drink (as well as its complete recipe) was mentioned on screen for the first time in the 2006 film adaptation of Casino Royale. The ' Vesper' became popular after the novel's publication.
In the novel, the character explains that she was born 'on a very stormy evening', and that her parents named her 'Vesper', Latin for 'evening' and Bond then gives her name to a cocktail he has recently invented.