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Recently, I've been thinking about constructing an argument consisting of either a tautology or something you can't attack, to make it more convincing.

In my opinion this is not allowed and I used to say "if you can use this argument anywhere else, it's not a compelling argument".

For example:

"X is wrong because, think about the children!"

Now my question is, is this really a fallacy (if so, what is its name?) or are there counter-examples?

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    I don't think it is a fallacy. For example, consider the argument "Raping children is wrong because, think about the children!" – alanf Jun 16 '14 at 10:07
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    Depending on context, it can be a fallacy, viz. appeal to emotion. But whether it is in any given argument fallacious would require details you haven't given us. – virmaior Jun 16 '14 at 15:22
  • Argumentum ad populum – Ron Royston Jun 22 '15 at 1:53
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When used as a plea for pity, this appeal to emotion can constitute a potential logical fallacy, while when used as an appeal for sympathy for weaker members of society, or the social good of the long-term health and viability of a society, it can constitute an argument for social justice generally accepted as appropriate.

[...]

Claiming to do something for the benefit of children is not a fallacy of itself, but if used to avoid logical debate, it is a thought-terminating cliché. Ethicist Jack Marshall described "think of the children!" as a "tried-and-true debate-stopper" used by "misty-eyed crusaders" to promote government policies and societal actions intended to alleviate the suffering or promote the well-being of children of improvident, impoverished, or otherwise flawed parents, which harms society by removing an important incentive for prudent behavior and by encouraging non-eugenic breeding by the type of person who has "fathered more kids than he can possibly support":

Wikipedia: "Think of the children"

"Think of the children!" is a tried-and-true debate-stopper, but more often than not one that succeeds because of its ability to inhibit rational thought. Children routinely have to suffer the consequences of adult incompetence, recklessness, stupidity, dishonesty and irresponsibility, and if preventing that biologically-dictated result is humanity's priority, then society needs to abolish the enforcement of laws, the obligation to support one's own family, and common sense. [...] [U]nless society sticks to principles that require adults to be responsible regarding the welfare of children in their charge, the "Think of the children!" reflex will suffocate order and justice.

Jack Marshall, "Think of the Children!": An Ethics Fallacy

I don't have access to it, but this may be interesting (and perhaps more scholarly) reading: Paul M. Pietroski, "Think of the Children", Australasian Journal of Philosophy, Volume 86, Issue 4, 2008.

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This might be a case of an "appeal to consequences" -- an unsound form of argumentation like "you should believe X because it will lead to good consequence Y".

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The most straightforward counterexample would be something like

Child abuse is wrong because, think about the children!

This is a rhetorical fallacy, but not a logical one. Objecting to any action because of its ultimate effects on anyone is not fallacious, but a causal argument of the effect or its unfairness is missing.

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