|Credit: see below|
Gdali Roisman brings to our attention the following suggestion 'for a new array'. It was published in Shachmat vol. 6 no. 3 (March 1967), p. 102. It was the suggestion of Meir Rom, and given the same reasons as other such suggestions, to wit, the "theory death" of chess.
Rom offers the new array tentatively, and suggests that players 'who are sick of playing by the book until the 20th move' will benefit from it. He notes the array doesn't change the rules (apart from the abolition of castling) or pieces, thus not requiring a change in middlegame or endgame strategies, or the production of new sets. The castling move is, of course, canceled in this array, but there are no other changes to the rules.
The array is interesting, being deliberately asymmetrical. presumably, to give play a more tactical, castling-on-opposite-sides character right from the start. The rooks and bishops, too, seem to start in a prima facie better, more active, position than in the orthodox game. This is unlike most suggestions for unorthodox starting positions, such as Fischer's "random chess" suggestion.
By the way, Tim Krabbe and others noted (though I cannot locate the exact place right now) that Fischer was hardly the first to suggest such a new array. Bronstein, Gulko, and others did, as well as Maarten van 't Kruijs in the 19th century, and the idea (as the article in the link notes) goes back further, to the 18th century. Van 't Kruijs was unorthodox in more ways than one: the "reverse French" (1. e3) is called the van 't Kruijs opening.
Such randomization comes readily to mind for those who wish to create a chess variant, as it requires no new rules, pieces, or boards, and is very easy to set up once the idea occurs to one. I won't be all surprised if Arabic (Shatranj) or even Indian chess was occasionally played in such a randomized fashion, for variety's sake.