Put your handlers in functions, to make it easy to refer to them in multiple places.
$(".aaa").on({
mouseenter: mouseEnter,
mouseleave: mouseLeave
});
function mouseEnter() {
$(this).css('background', '#dddddd');
}
function mouseLeave() {
$(this).css('background', '#888888');
}
$(".aaa").click(function() {
$(this).css('background', '#555555');
$(".aaa").not(this).css('background', '#111111');
$(this).off('mouseenter mouseleave');
$(".aaa").not(this).on({
mouseenter: mouseEnter,
mouseleave: mouseLeave
});
});