Kasparov proposed one idea in an interview he gave shortly after he retired from chess in 2005: Garry Kasparov Interview, Part 2 (part 1 is linked at the end of the second part). The 13th World Champion said,
I’m in favor of at least investigating doing one position per year from chess960. I know the reaction is "Aahhh, horrible!" Most players think it’s terrible, saying the purpose is to start each game from scratch. I don’t think so, I think the point is to create more space for creativity. If you have a position for a year you can’t go too far in analysis. You can reach move maybe four or five, that’s a lot of room for creativity.
The purpose of the suggestion seems to be letting professional players continue with the working methods they have used for decades -- preparing opening systems in advance, committing them to memory, and playing their ideas in important games against unsuspecting opponents. It has the advantage of keeping an element of psychology in the game.
The lack of opening psychology is a noticeable disadvantage of chess960. In traditional chess a player makes a statement in playing 1.d4 instead of 1.e4, in playing the Caro Kann instead of the Sicilian, in choosing a rare side variation instead of a main line, and so on. The psychological struggle in the opening is even more important for players who meet each other regularly, like the world's elite do.
Unfortunately, a rhythm of one position per year wouldn't have much impact on the publishing business. A year is not enough time to gather experience with a particular start position, to write a book on that experience, to publish and market the book, and to use the ideas contained in the book in further competition.
A good starting point for Kasparov's proposal would be one of the 11 symmetric positions (12 positions minus RNBQKBNR) with the King and Queen on the central files that I listed in A Database of Chess960 Start Positions; for example, RNBKQBNR (SP534). Since it is the closest relative to the traditional start position (SP518), many of the ideas known to current chess theory would be naturally reused. It would also show what impact the slight change in the castling rules, described in Arbitrariness in Chess Rules, has on the overall game. This would be useful knowledge for evaluating differences between the other 479 chess960 twins.
In one of his radio interviews, Fischer scoffed at Kasparov's idea. Perhaps he reacted too hastily.
Later: A better start position to explore Kasparov's idea might be what H.J.R.Murray (see More on the Initial Position in Chess) called the 'crosswise arrangement: where the King and Queen start not on the same file, but where the King starts on the same file as the opposing Queen'. This is not a chess960 position, but still has the advantage of being unexplored opening territory.