What observational data convinced Stephen Hawking that Cygnus X-1 was indeed a black hole and caused him to break into Kip Thorne's office? It's generally easier to be certain an object is a star than a black hole. HDE 226868 is one of our site moderators and a blue supergiant star. The latter has a 5.6 day companion that supplied a magazine subscription to Kip Thorne:


  According to the updated tenth-anniversary edition of A Brief History of Time, Hawking has conceded the bet due to subsequent observational data in favor of black holes. In his own book, Black Holes and Time Warps, Thorne reports that Hawking conceded the bet by breaking into Thorne's office while he was in Russia, finding the framed bet, and signing it. (While Hawking referred to the bet as taking place in 1975, the written bet itself (in Thorne's handwriting, with his and Hawking's signatures) bears additional witness signatures under a legend stating "Witnessed this tenth day of December 1974". This date was confirmed by Kip Thorne on the January 10, 2018 episode of Nova on PBS.)


Question: What observational data convinced Stephen Hawking that Cygnus X-1 was indeed a black hole and caused him to break into Kip Thorne's office?
From The Unknown Universe: What We Don't Know About Time and Space in Ten Chapters  by Stuart Clark:


  By 1998, Hawking was ready to concede the bet. He thought that although the specific evidence for Cygnus X-1 had not changed, there was now so much evidence for other black holes dotted around the galaxy an the wider Universe that there was no more room for reasonable doubt. Besides, [...], he had bigger fish to fry and a new bet to concentrate on. So Hawking paid for Thorne's prize. 


Although Kip Thorne himself apparently still had some doubts until 2011  [see here]: 


  "For forty years, Cygnus X-1 has been the iconic example of a black hole. However, despite Hawking's concession, I have never been completely convinced that it really does contain a black hole - until now," said Thorne. "The data and modeling described in these three papers at last provide a completely definitive description of this binary system."


The three paper he is referring to are: 


Reid et al., “The Trigonometric Parallax of Cygnus X-1”   
Orosz et al., “The Mass of the Black Hole in Cygnus X-1”  
Gou et al., “The Extreme Spin of the Black Hole in Cygnus X-1”