Term For Argument by Showing 1 Counter Example

I'd call this a case of anecdotal fallacy.

Anecdotal fallacy – using a personal experience or an isolated example instead of sound reasoning or compelling evidence.

A philosophical term also known as misleading vividness is anecdotal evidence describing an occurrence with sufficient detail to permit hasty generalizations about the occurrence. It may be used, for example, to convince someone that the occurrence is a widespread problem. Although misleading vividness does little to support an argument logically, it can have a very strong psychological effect because of a cognitive heuristic called the availability heuristic.

Example:

Anne:
"I am giving up extreme sports now that I have children. I think I will take up golf."

Bill:
"I wouldn't do that. Do you remember Charles? He was playing golf when he got hit by a golf-cart. It broke his leg, and he fell over, giving himself a concussion. He was in hospital for a week and still walks with a limp. I would stick to paragliding!"

References:
https://en.wikipedia.org/wiki/List_of_fallacies
https://en.wikipedia.org/wiki/Misleading_vividness

I think you are referring to the: "Exception that proves the rule":

• The exception [that] proves the rule” is a saying whose meaning has been interpreted or misinterpreted in various ways. Its true, or at least original, meaning is that the presence of an exception applying to a specific case establishes (“proves”) that a general rule exists. For example, a sign that says “parking prohibited on Sundays” (the exception) “proves” that parking is allowed on the other six days of the week (the rule). A more explicit phrasing might be “the exception that proves the existence of the rule.”

(Wikipedia)