Mediation can work at any stage of a dispute -- even after trial. But so very often the best time to mediate a conflict is before litigation is commenced. Before, I would argue, a complaint is even filed 
Psychology students will tell you of the well-known race track study which demonstrated that bettors were more confident of their pick moments after placing their bets than they were right before laying their money down at the window.
Sales professionals repeat the (probably apocryphal) story that toy manufactures hit on a technique for doubling their sales by under supplying demand. Before Christmas they create a demand for an item, and then short retailers while simultaneously making a near-substitute readily available. In January, they unleash the under-supplied toy and parents everywhere rush to fulfill their pre-Holiday promises to the litte ones. Voila! Two sales instead of one.
So what do horse races and fad toys have to do with mediation? They illustrate the legitimate psychological barriers to resolution that mediators need to address, particularly in late-stage mediations -- after all the discovery, demands, accusations, drawn lines have been made.
Even before a lawsuit starts, attorneys advise their clients on what they are going to go after, or how they will cleverly fend off the plaintiff; case valuations are declared, expectations are set. Promises are made, or at least tacitly understood. And when it comes time to sit down with a mediator, all of that history rubs against the natural human impulses of commitment and consistency -- the commitment of the horse bettor to the cash laid down, the commitment of the attorney to the righteousness of the cause; the promise of the parent to feed the fad, the promise of the litigator to bring home a certain result.
Techniques for dealing with consistency and commitment issues are part of the training of mediators. The most important lesson, though, is that the issue is not one of ego, or saving face. Following through on commitment and promises are fundamental human urges (we'll leave it to evolutionary psychologists to say where they came from).
Understood that way, the mediator realizes that the task at hand is to provide a neutral, safe environment for reconsideration. Often I will avoid restatement of well-know positions and claims in a late-stage mediation and move to private meetings for those discussions -- this seems to facilitate the ability of the parties to separate from the hardened positions.
But more to the point of this piece -- the sooner a conflict is brought to mediation in the first place, the easier it is for the parties to focus on interest, needs, and remedies, as opposed to the nag in the fifth or this year's Teddy Ruxpin.