Wishes are a copout in the original game as well as in fiction. Wishes come into play when there is nothing left, and even the magic rules are inadequate to the situation.
They're kind of the equivalent of "And with one mighty bound, he was free!" or Asimov's dreaded Pocket Franistan (I hear that was in an auction in Sotheby's a couple of years back. Who's got it now?) - they're a way of getting yourselves out of a corner you, or the ref, have painted yourselves into.
Wishes are, in effect, a call for Divine Intervention - an invocation of a Higher Power to come in. A bit like a kid calling out for its Mummy or Daddy under duress. "Author, O Author, please please please rewrite this hideous storyline I've placed myself into!"
(Ten years ago, I'd have said "Please, Jim, can you fix it for me ..." *shudders*)
The mechanism can be worked out easily in science fiction terms.
The Artefact is a device which permits retroactive quantum manipulation of timelines. It detects high levels of stress - anger, fear, pain, desire, imminent death - and it goes back a few save levels, as it were, back to a previous time where the character was faced with a decision. The Artefact then alters the timeline so that an alternate decision was made, leading to a more favourable outcome in the present day.
Example: Captain Rosphert McAbrin of Free Trader McGinty is running short on laser pistol charges, and there seem to be no end to the aliens storming his barricade seeking his head on a plate. He invokes the Artefact's help, and as he steps back his foot bumps against a bag that he could have sworn that he'd left behind back in the Ship's Locker. Then he remembers differently, and realises that he had, indeed, taken the bag despite the risk of encumbrance. A bag loaded with spare laser pistols and several fully charged clips.
There is one small problem with the device. With wishes, there are always problems. Either the wish device is limited in its number of uses before it moves on to the next person, or each activation sends the wearer into a parallel universe in a different timestream to his original timeline. One where, despite the Artefact keeping the spacetime distortion as localised as possible, there is still leakage, alteration of history and consequences.
Example: Captain Rosphert McAbrin returns to the spaceport, only to find that his First Officer is dead. Agnes ran short of charges in her laser pistol. Aliens got her and killed her. In the previous iteration of time, she had been the one to take the holdall containing the extra charges and she survived. Now the Captain has to mourn the loss of not only a fine First Officer, but also his girlfriend.
Like time travel stories, there is always a consequence to rewriting history or rewriting the local laws of physics. And if you look closely, you'll see that the consequence is almost always some form of tragedy.
In Charmed, there was a huge restriction in the way magic could be used; the Halliwell Sisters were forbidden to use their powers for what was termed "personal gain." They couldn't conjure up a man for their lives, or magic away the dirt so they didn't have to put in the elbow grease and do the laundry. Not without hideous consequences, usually in the form of some squirmworthy humiliation, such as the man conjured up turning out to be gay, or the dirt turning into some sort of an Evil Dirt Monster and mucking up the town or something.
So to put things right, there always has to be some form of sacrifice. Kirk loses Edith Keeler, and later Spock loses his sehlat, on passing through the Guardian of Forever; and when the poorly-named Captain Annorax of the Krenim tries to work the timelines, the first temporal alteration he makes causes his wife to cease to exist in his timeline and in all the timelines. It is only when he is destroyed that time returns to normal, and he gets to be with his wife again.
The ref can work that into the functioning of the device.