Oh but this is just an exception

Oh, ain't this common. And it could be multiple fallacies (or not fallacious at all) depending on context and whether the exception is justified.

special pleading is one possible fallacy. This applies when someone creates a rule or principle, then excuses themselves from it, without a proper reason. An example would be a mother who believes all drink drivers who kill someone should go to prison, who then says that her son - a convicted drink driver who killed someone - shouldn't go to prison because "he's actually a really good boy who made a terrible mistake."

double standard an the application of special pleading 

If it's not a fallacy, this is because the exception is valid.

Whether the exception holds depends on context and consequences. There's no absolute standard, though we can say broadly that two things should be treated equally unless there is a rational distinction that can be made between them, which would justify one being treated differently. Thus, you'd have different standards, not double standards.

Why take this "For every rule there is an exception and this is the exception to the rule." as a given?   i.e. if there is an exception to the 'rule', then perhaps the rule was not a rule at all, i.e. in hindsight, it was ambiguously defined, or an evolving understanding of the 'rule', perhaps a 'soft' rule, as is pointed out in the reference to statistics, perhaps even a 'rule of thumb'.

Perhaps we can show "For every rule there is an exception and this is the exception to the rule." to be invalid?  Think of a 'rule', a true, rule without exception, perhaps a boolean rule, or a math.  Say, if A=B, then B=A.  There is no exception to this rule, yes?   So "For every rule there is an exception" is shown false, yes?  I am not sure what fallacy would be invoked here, but I do think the premise is invalid.

This is basically no true scotsman , where the arguer would respond to a purported counter-example by saying the counter-example doesn't really apply.  

It's also a self-sealing argument , which is a form of unfalsifiability,  in that no matter what evidence the responding party comes up with, the arguer dismisses it as not being a real counter-example to their claim.

This is an amended answer to the question, clarifying my earlier answer.

I think this is a variant of no true scotsman.  The basic form is:

1) Arguer claims:        All P are Q.  (This is a generalization, whether it applies to a rule or a group).

2) Responder:            But what about X?   X is an example of P for which Q doesn’t follow.

3) Arguer responds:   X isn’t a true example of P.   

C).      Therefore, the generalization still holds:   All (true) P are Q.

The variant suggested by the Question might be called “That’s an Exception Fallacy”.   1) and 2) and the Conclusion C) are the same but the Arguer's reply is different:

1) Arguer claims:     All P are Q.  

2) Responder:         But what about X?   X is an example of P for which Q doesn’t follow.

3) Arguer responds:   You're right; X is an example of P for which Q doesn't follow, but X is an exception.

C).      Therefore, the generalization still holds.   All (true) P are Q.

Somehow, by calling something an "exception", we’re supposed to ignore the fact that the general rule doesn’t always apply.  This is committing several fallacies including what’s called the Nominal Fallacy, the fallacy of assuming that just because you’ve given something a name (in this case “an exception”)  that you’ve actually said something of substance.  

 In both cases, the retort by the arguer is committing the unfalsifiability fallacy; any exception to the general rule is explained away--either as not genuine or as an exception, so there is no way of falsifying the general rule.

Note: On the other hand, to insist that there be no exceptions to a rule even when the rule wasn’t reasonably intended to have no exceptions, commits a different fallacy: accident fallacy 

In my experience, people use the cliché it is the exception that proves the rule, or a variant, in casual conversation. But, even then, it is a smokescreen used to cover the fact that either proposition is imprecisely stated or the reasoning is flawed. 

The correct response is to investigate the matter more carefully and restate the proposition. 

Claiming something is an exception to a rule is basically claiming that an event is statistically insignificant. This is either true or false. For example, the smoker who lives to 105 is the exception to the rule that smokers generally die 20 year sooner than their non-smoking counterpart. This is valid exception because the data supports it.

There is no strict demarcation for what qualifies as an exception. Using my same example, a smoker who lived until 80 might be an "exception to the rule," but more likely just on the edge of the statistical curve in the normal distribution. Of course, if someone claimed that rule that doesn't exist and the typical case was an exception, they would also be wrong (i.e, making a false claim).

I see no fallacy here.