This isn't necessarily related inherently to randomness, but more to the last point in the article about remedying wrongs and the general point about consistency. In running a game of Legend of the Five Rings--a system where the level of success of an action is dependent on player decision before the roll on how awesome they're trying to be, rather than watching the die and waiting for a 20--I ruled in the first session that a player character had essentially a critical success due to an absurdly high roll result. So awesome that we basically got him an Ally advantage to follow with the nature of the success I described. Normally, advantages can't be bought after character creation, but hey, it was the first session, and he had the experience unspent to devote to it anyway, no big deal.
And then I remembered the part about the height of a roll not determining level of success. Oops.
Thus was instantiated the Max Advantage, a one-shot bonus named after the aforementioned player, which every character in the game received at the beginning of the next session. Anytime someone happens to roll super-high and didn't declare an attempt for an outstanding result, they can blow their Max Advantage to get a spectacular success and the chance to purchase a character-sheet advantage of my selection based on the situation.
So far we've had two other players pop their Max Advantage, one gaining a bloodline Advantage for the Fortune of Thunder, since his spectacular success was on a spellcasting roll the first time he cast Fury of Osano-Wo, a flashy lightning spell. Another one used his Max Advantage to cleanly cut off a demon's arm (and bathe one of its allies in its corrupted blood, knocking him prone), and he bought the Sacred Weapon advantage to receive a way better katana next time he was in front of his lord.
Way better than simply setting things right, this oh-no moment early in the game has made things significantly more fun in the long run, and has given me something long-term to express a sense of longevity in the campaign. And it's also given me an out for other times I screw up with the system--we have a Travis advantage, too, though it's much less interesting for the purposes of this discussion.
Otherwise on the topic, I shall just note that I also GM in a system where the book advocates that you should fudge the rolls any time (and every time!) that it makes the result more fun.