According to John Byrne, the inspiration for the Hellfire Club came from the UK television show The Avengers and specifically the episode titled A Touch of Brimstone (February 1966). That episode also featured actor Peter Wyngard, who served as the model for Jason Wyngard. The name Jason came from a later series Wyngarde starred in, Jason King, which was itself spun off from a show called Department S.
In A Touch of Brimstone, Avengers John Steed and Emma Peel investigate a secret organization comprised of London's wealthy elite that are plotting economic and political terrorism in the United Kingdom. Having watched the episode recently for the first time, it's clear how well the show served Claremont and Byrne as a source of inspiration. It's really a great blast to the past if you've never seen the episode.
Easily the most memorable part of A Touch of Brimstone is Emma Peel's dazzling transformation into the Queen of Sin during her infiltration of the Hellfire Club. Her black bustier, leather boots, and choker were clearly the inspiration for Jean Grey's Black Queen ensemble and were definitely very racy fare for 1966. So racy, in fact, that the episode was initially not allowed to be broadcast in the United States.
While A Touch of Brimstone clearly served as inspiration to the creative team in many ways, Claremont and Byrne breathed their own life into the X-Men's Hellfire Club and created a group of villains that has been one of the X-Men's most enduring and relevant foes for more than three decades and counting. Their work has definitely withstood the test of time and the Dark Phoenix Saga is as gripping today as it ever was!
John Byrne Forum, The Avengers Forever