The problem, by one analysis, is that patients have no incentive to weigh costs against benefits. That's a tough task in any case, and pushing it off to the insurers makes it harder, as the insurer has less information than either the patient or their doctor about their individual case. The result is that neither patients nor (often) doctors know what the services cost.
In fact price information has been so effectively removed from the system that in some cases nobody knows what the service really costs. tptacek posted a great example: http://news.ycombinator.com/item?id=4465845
The purpose of the mandatory minimum copay is to force price information back into the system. When patients have to pay some part of the price of every service, they will therefore have to know what those prices are.
All that said, there are obviously some questions and problems to be dealt with. For starters, what's the formula for how much people have to pay? On the large end, I would suggest something like (price in dollars) ^ 0.7. This means for a $10k service, the copay would be about $600; for a $100k service, about $3000; and for a $1M service, around $16k. Those numbers feel reasonable to me: big enough to make most people think twice about choosing a six-figure intervention, but not an absolute barrier if there's a good medical reason.
On the small end, though, that formula produces numbers that are arguably too large. A $200 service (if there is such a thing? maybe in lower-income rural areas) would have a $40 copay by that formula. This risks discouraging people from getting preventative and urgent care that they need. To correct for this we could simply subtract some fixed number from the result of the above formula. I'm not sure what the right number is, but somewhere in the $40 - $80 range, I would guess. (This provision could also be subject to means testing.)
Probably when I say "mandatory" what I really mean is "the government provides incentives to make plans with this property significantly cheaper than those without". I don't care that rich people will circumvent the rule if they want; I just care that most people are on plans with this property, so that, as I say, price information is forced to flow again.