Here are a few things that have recently been called "the exception that proves the rule":
1. The Mini-Transat sailing race (because it's international and doesn't "boil down to duels between French sailors")
2. Adrian Peterson (because his success as a football player is not about "opportunity and scheme" but unusual talent)
3. A congressional compromise on student loans (because Congress reached a compromise!)
Taken as a set phrase, "the exception that proves the rule" indicates a deviation from the norm, a challenge to the stereotype. It says, in effect, the norm or stereotype is the rule and here is something that is an exception to that rule. But wait, how does the exception prove the rule? Wouldn't it do just the opposite? Doesn't it prove that the rule does not hold for all cases and is therefore not a rule at all?
It is sometimes argued that the confusion over this expression stems from the wrong understanding of "prove," that "prove" here means "test," as in "proving ground" or a printer's proof. The idea is that the exception tests the validity of the rule, and that test could either leave the rule intact (if some kind of explanation can be found) or overturn it. However, it's hard to come up with an example where that is truly what is intended by the phrase. It almost always carries the assumption that the rule remains intact.
In fact, the "prove" part of the phrase was not very important in its original formulation. The expression comes from the Latin legal principle exceptio probat regulam (the exception proves the rule), also rendered as exceptio firmat regulam (the exception establishes the rule) and exceptio confirmat regulam (the exception confirms the rule). The principle provides legal cover for inferences such as the following: If I see a sign reading "no swimming allowed after 10pm," I can assume swimming is allowed before that time; if an appliance store says "pre-paid delivery required for refrigerators," I can assume they do not require pre-paid delivery for other items. The exception here is not a thing but an act of excepting.
The act of stipulating a condition for when something is disallowed (or required), proves that when the stipulated conditions do not hold, it is allowed (or not required). The general rules are that swimming is allowed before 10pm and that pre-paid delivery is not required. The fact that exceptions to those rules have been stated confirms those rules hold in all other cases. The full statement of the principle reads exceptio probat regulam, in casibus non exceptis. The exception proves the rule in cases not excepted.
These days, you'd have to be one heck of a stickler to insist that "the exception that proves the rule" only be used in its original Latin sense. It has probably come to its current sense through some blending with the expression, "every rule has an exception." Most of the time, when people say something is "the exception that proves the rule," they could just as well say it's "an exception to the rule."
I would argue, however, that "the exception that proves the rule" does more by highlighting the unusualness of the exception. The Mini-Transat, or Adrian Peterson, or a Congress reaching an agreement are not just outside the norm, they are so far outside the norm they force you to notice what the norm is, or that there is one at all. There is an important kernel of the original sense here. The existence of the exception gives force to the rule. The no swimming after 10pm sign makes it clear that it's okay to swim before that. The congressional compromise on student loans makes it all the more clear that congress can't compromise on anything else. Sure, there's usually a good bit of hyperbole going on when someone pulls out this expression, but that doesn't mean it doesn't make any sense.