The familiar, quick-tempo song heard when the children are making a mad dash to the toy store to buy the Hula-Hoops is called "Sabre Dance". It was written by Aram Khachaturyan and is featured in his ballet, "Gayane". The song is often associated with juggling acts and such.
The published script is introduced with a fake interview with "producer" Joel Silver (who is uncredited in the film). In this interview Silver claims that the Coens were difficult to work with and among other things wanted Ethan to play Tim Robbins's part and Jeanne Moreau to play Jennifer Jason Leigh's (despite of the age difference of 30+ years).
The main theme music of the film is the uncredited Adagio of Spartacus and Phrygia from the ballet Spartacus by Aram Khachaturyan. This music was also used as the main theme for BBC TV's series The Onedin Line (1971). The music played behind the black and white newsreel is Non-Stop by John Malcolm, used by Independent Television News in the UK in the 1950s and 1960s to introduce their main news bulletins. Malcolm apparently composed the piece to blow a raspberry at the musical prejudices of his tutor.
The secretary of the Hudsucker brainstormers (uncredited Mary Beth Peil) is reading throughout the Hula Hoop manufacturing sequence the novels "War & Peace" and "Anna Karenina" by Leo Tolstoy. The idea is to show that the brainstormers are really taking their time to determine the name of the product since the novels have, respectively, 1225 and 864 pages and are commonly known for being books you would spend a lot of time reading. In her case, if you'd consider 4 minutes per page, 8 hours a day, both books would've taken 18 days to read.
The trivia item below may give away important plot points.
The death of Waring Hudsucker was inspired by a real-life incident. On February 3, 1975, Eli Black, the CEO of the United Fruit Company, smashed an office window with his briefcase and jumped to his death from the 44th floor of the Pan Am Building in New York City.