Is there a widely accepted name for this fallacy?

The fallacy I’m thinking of is the one where an argument is attacked based on its perceived weakest link, while failing to address other, potentially persuasive elements.

An example is when a debater/poster gives a number of examples to illustrate a proposition, and others jump in to challenge one specific example and give the impression that refuting that example invalidates the entire proposition. This happens not infrequently on the Dope, and I have probably been guilty of it myself.

A cite from the world of sports (relating to lawsuits against Texans quarterback Deshaun Watson) may be relevant. From ESPN dot com:

"Watson’s attorney, Rusty Hardin, said…in a statement last week that his law firm has “strong evidence” showing that one of the lawsuits alleging sexual assault is false and that it “calls into question the legitimacy of the other cases as well.”

I don’t want to get into a debate about the Watson affair, but one can doubt whether debunking a single accuser’s claims invalidates what’s alleged in all the other cases (there are now a total of 19 lawsuits).

The nearest I can find is the nut-picking fallacy, where someone seizes on the actions of a single person (generally a fringe character) and pretends that they are characteristic of an entire group.

Is there a “weak link” fallacy?

I’m not sure its a fallacy. Its certainly a mainstay of attacking the opposing case in a trial. If you can show that one piece of an opposing case is false, then it undermines the whole thing (“If the glove don’t fit, you must acquit” being the most famous example).

But this is multiple cases, not one.

Anyway, at the OP: I think it might just be plain cherry-picking, no other name.

I don’t know if there’s a common name for it, but it’s anti-steelmanning.

Steelmanning is taking the strongest possible version of an opponent’s argument, and arguing against that. It’s a form of the principle of charity, i.e., that you should be charitable towards your opponent and not simply pick on the weak points.

Steelmanning (even in the definition link above) is sometimes said to be the opposite of strawmanning, but that’s not really correct. Strawmanning is a misrepresentation of the opponent’s argument in a weaker form (often this is just taking else’s argument and arguing against that). Anti-steelmanning is what you described above: arguing against the weakest part of the opponent’s argument instead of the strongest part.

I’d say it is the “falsus in uno, falsus in omnibus” fallacy, except that isn’t really a fallacy so much as it is a flawed truism. It is a principle of law that if you can show one falsehood in testimony, then the entire testimony is suspect, but we all understand that’s not really true. If I make a minor demonstrable error in an hour of testimony, I’ll call that a goof while the other side’s attorney may call that invalidating every word I said.

When you say “weakest link” you make me think you’re talking about logically linked statements in a chain of reasoning. In that case it would be wholly appropriate to attack an argument at its weakest link.

But then you talk about challenging one specific example among several, which is a different kind of situation.

It’s the difference between the elements of the argument working in serial vs. in parallel.

If the argument has multiple pieces of evidence working in parallel, then yes, challenging one element doesn’t invalidate the argument as long as there are enough other strong elements.

But because people do this, it’s probably not a good idea to include one element or piece of evidence that’s much weaker than the rest, if you’re the one making the argument.

Sounds like a variation on the Encyclopedia Brown method, a.k.a. Conviction by Contradiction.

I have heard it called “weak man argument”, a sub category of straw man.

In addition to what others have said, it’s often just a means of obfuscating.

In other words, people often make a fuss about their strongest point (your weakest point) in the hope that you won’t notice they aren’t dealing with your other points (or that onlookers won’t).

I think I’ll refer to it as the bad apple fallacy.