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In relation to my question on Mathematics.SE, I would like to know what English words would fit the best to describe the expected amount of information that will be gained by learning the answer to a given question or by learning the responses to a given questionnaire.

The word should describe the question or the questionnaire, not the answers. Thus, words like "informativeness" or "information value" would fit poorly, because the common view is that the information is contained in answers, not in questions (though in fact both are important).

For example, the value that is to be described is a priori higher for the question "What is your favourite dish?" than for the question "Do you like French fries?" It is also a priori higher for the question "Are you married?" than for the questions "Do you pay taxes?" and "Are you a lawyer?" (assuming nothing is known about the person being asked, the answers to the last two questions are easier to guess in advance -- "yes" and "no" respectively).

Usage examples:

The question "What is your favourite dish?" has higher foo (or more foo) than the question "Do you like French fries?"

The total foo of a questionnaire is the sum of foos of its questions.

I know that this is an instance of entropy value, but I am looking for a more common and intuitive term, if such exists.

You mentioned the technical term

entropy

which means, in the information theory sense, the number of bits of information in a message, or likewise the thing you are asking for. It is however pretty technical, and also has its provably equivalent but equally obscure interpretation in thermodynamics. As such, replacing the informal phrase 'amount of information' with entropy will be only understood under very technical contexts.

You may be looking for an informal synonym, one that can replace 'entropy' to get the same idea, analogous to your example of 'salt' for 'sodium chloride'. As 'entropy' is a latter day technical invention for a new concept (by Rudolf Clausius in 1854), there is no existing informal older term.

So, there is no single informal word for entropy.

That's a hard thing to prove definitively (as opposed to suggesting a term that every one can see and judge yes or no). But the timing of the term is some justification that it is unlikely.

Asking for a single word for a complex concept is expecting a lot of any language. English and more likely its fans may have spoiled you by popularizing all the ineffable situations that turn out to have single words for them (eg 'justice', 'game', 'ineffable')

However, there may well be near enough synonyms or synonymous phrases for the concept. The title 'amount of information' is sufficient or even simply

'information'.

One question can be a better discriminator than another.

A characteristic which enables people or things to be distinguished from one another.

[Oxford]

Although the question does not contain the sought-after information, one question has the ability to elicit more information than another. You will find references in the psychological testing domain to high-information yielding questions as good discriminators. For instance, if a particular math problem covaries meaningfully with IQ, that problem is said to be a good discriminator for IQ.

If I'm trying to find a criminal suspect who is an American male, the question "Is he over 7 feet" is a poor discriminator, because it provides very little information for the answer "no". An If I use the median of the population of 5 feet, 9.5 inches, the question is guaranteed to eliminate half the population, which is the maximum one can guarantee, regardless of the answer.