"Had better" is used to strongly suggest a particular course of action. While it can be included in questions, the only typical use would be suggestions phrased as questions:
Hadn't you better get to school? It's already almost time for your class.
The speaker is not really asking a question here. It's simply phrased as a question because this can sound more polite. This is why the responses to these questions are normally either:
Yes, I had (because ...)
No, I don't have to (because ...)
In this it's not really different from other questions that ask for confirmation:
Weren't you late to class yesterday?
Didn't you wash your face after you woke up?
However, in some contexts "had better" works the same as "should" or "ought". So it is possible to use "had better" in an information question, as in your example:
What is something you had better do if you've lost your child in the mall?
but, at least to me, this still sounds slightly odd. For clarity I would instead say:
What is something you should do if you've lost your child in the mall?
or even better:
What should you do if you've lost your child in the mall?