It is generally understood that White has a slight advantage, by virtue of moving first. To offset that advantage, and put Black on a more equal footing, I propose that Black be allowed to castle out of check.
This change would ease somewhat the pressure Black is under in the opening, in which he is trying simultaneously to develop his pieces and attend to the safety of his King. A check by White can force Black to relinquish his right to castle, seriously compromising his game. Allowing Black to castle out of check would enable him to parry White's developing moves less passively, resulting in a game with more equal chances for either side to win.
In addition to making the game more even, this variant would present a vast new territory of opening theory for Chess players to explore.