The ChessVariants.com page on Fischer Random Chess says,
Fischer Random Chess is somewhat similar to the older Shuffle Chess, or Prechess (or other related variants), yet has a unique style of its own. An extensive introduction and history of the game was written by Eric van Reem.
The van Reem article, The birth of Fischer Random Chess, is available on multiple web sites. It dates the 'birth' to Buenos Aires 1996, the event that I covered in Fischer Announces Fischerandom. Gligoric, in his ground-breaking 'Shall We Play Fischerandom Chess?', gave a more detailed account. After mentioning Capablanca's erstwhile efforts to improve the game that he felt was under threat from draws, Gligoric wrote,
Fischer's concentration on this problem went much deeper. During his stay in Saint Stefan [better known outside Serbia as Sveti Stefan] in 1992, he recommended shuffling all the pieces at random on the back row before the beginning of each game. [...]
It turned out that Fischer's first plan would make 2400 different commencing positions. Immediately after his sensational return to the chess scene in 1992, he began experimenting privately with this kind of chess against colleagues and chance visitors. Although Fischer was pleased that the mathematical sum of starting positions was very large, he soon discovered that eventually having two Bishops of the same color made an unpleasant impression, producing one-sided and limited opportunities on the chessboard. It was also clear that obstructing the right to castle would mean a step backward toward the primitive distant chess past and if the intermingling of pieces on the back row made castling impossible for both sides, this would inflict irreparable damage on playing strategy as well.
The result of Fischer's constant meditation on how to give alternative life to the game of chess, threatened by the exhaustion of its creative resources, was the formulation, in September 1993, of the rules of "Fischerandom Chess" which are quoted in a separate chapter of this book. [p.36]
I covered those rules in Fischer's Rules of Fischerandom.
Prechess, mentioned above in the ChessVariants.com quote, is a variant of shuffle chess where the players take turns placing single pieces on the back rank. Since the rules don't appear to have been formalized anywhere, restrictions on the Bishops and possibilities for castling depend on who is playing and what was agreed beforehand.
Who is the 'Father of Chess960'? There are three key requirements: (1) shuffle chess with the White and Black pieces starting opposite each other; (2) the two Bishops for each side on different colored squares; and (3) castling into the familiar O-O and O-O-O positions. Other would-be inventors might have proposed (1) and (2), but Fischer proposed (3). That last contribution, which steers the game from a shuffled variant into positions fully reminiscent of traditional chess, gives Fischer the title.