Argument by Exception

Is there a fallacy for arguing against something by citing exceptional examples? I get this all the time at work when people shoot down a broadly sound idea by citing exceptional examples.

E.g. someone might make the claim that lower skilled people earn less money. Someone replies that, actually did you know that plumbers earn more than lawyers! These exceptional examples might be true in the exception but they are used in this rhetorical way to neutralize the claim on-the-whole.

Another way I hear it used is to argue against a general theory - e.g. when I'm talking to my racist uncle:
Me: black people are disadvantaged
Uncle: no they're not, they can do just as well as whites - look at Barack Obama!

There is some ambiguity going on here that gets in the way of mutual understanding. When someone makes a general statement they can either mean "all," "most," or "some." Unless these words are used, the person is being unclear. Consider the different meanings of the following:

All lower skilled people earn less money (then higher skilled people).
Most lower skilled people earn less money (then higher skilled people).
Some lower skilled people earn less money (then higher skilled people).

Just saying "lower skilled people earn less money" could prompt someone to reasonably respond with a counter example to that claim, so that would not be fallacious. If "most" or "some" were used, then the counter example would be a non sequitur , as the "objection" wouldn't follow.

Your second example is a bit different, let's see why:

You: black people are disadvantaged
(general statement, not completely clear, but can reasonably be understood to mean "on average").

Uncle: no they're not, they can do just as well as whites - look at Barack Obama!

Here, your uncle is answering your general claim (understanding that you mean "on average") but using one specific example ( anecdotal fallacy ).
Had he said, "Certainly not all black people, look at Barack Obama," then you might argue that all black people are disadvantaged, but that doesn't mean they still can't be successful or you can clarify and say "I meant on average. Surely some have more advantages than some white people."

So in summary, it is best to be clear when making general statements as to what it is precisely that you mean. The responses with specific examples that counter a broad statement might be okay, but it depends on how it is phrased.

Skilled just means trained and/or qualified to a certain level, and I'm not sure that lawyers are any more skilled than plumbers.

Also lawyer conjures up an image of high paid defense attorneys or barristers, while the majority of lawyers probably spend their time working in offices writing documents. The former group are more highly paid, and even those with higher ability (often in rhetoric or even just memory) command considerably more than a plumber.

So I think you could say that's a fallacy of vagueness regarding the term lawyer, and possibly a fallacy of equivocation regarding the term skilled. You could possibly even say that's a straw man as you said lower skilled, and to me semi-skilled or unskilled is lower than skilled, and the reply compared skilled with skilled.

With regard to your uncle, perhaps he's not aware that golfers who play off scratch can beat people who are given strokes. Having an advantage or disadvantage doesn't determine someone's success in an endeavour