What attracts people to learning things that they consider neither interesting nor important?

There's a risk of using status considerations as a hammer against which everything looks like a nail, but in this context I really do think they're pretty often the right answer. Amy Chua, for example, wasn't forcing her daughters to learn violin or piano (and only those instruments, IIRC) primarily because it builds some vague form of character; she was doing it, consciously or not, because violin and piano are coded as upper-class instruments and learning them is a form of costly signaling. Contrast, say, guitar; becoming a competent guitarist isn't much easier than becoming a competent pianist (I tried both when I was younger, without much success), and any cognitive effects from learning it must be much the same, but it lacks the desired class and personality associations.

Are they consciously aware that they find what they're learning neither interesting or important? I think I've met quite a few people who studied things because they believed themselves to be interested in them even though you could infer from their behaviour that actually they weren't. We often fail to know our own preferences.

Can you give a few examples where people voluntarily learn things they consider neither interesting nor important? I am having trouble thinking of any.

Reason number one: someone with power or influence over them considers the activity to be interesting or important. (Example: school.)

I'm not sure why you're dismissing building character as an explanation. Something builds character if it helps a person develop virtues such as patience, perseverance, humility, temperance, etc. Committing to a difficult activity can obviously do this, perhaps more so if it is not instrumental. There's also the sense in which an activity can be a test of character, so that completing it reveals (to oneself and others) virtues (or room for improvement). I find "direct hedonic value" far more suspicious, since most "rewarding" activities offer nothing of the sort (the pleasure of a difficult activity is usually delayed and the activity itself may be wholly negative), and those that do are usually considered addictions/vices/etc. Hedonic reward also has only a tenuous connection to an activity being considered "interesting"; in fact, probably only boredom can stop an activity from being interesting and I'm not even sure about that (in many endurance sports the tedium is part of the challenge).

It seems obvious to me that there are cases (not only involving learning) in which people quite reasonably do things neither for your reason 1 nor for your reason 2. For instance, people often do things in order to be paid, even though they don't enjoy them all that much for their own sake and they don't do much to build skills. People do things to impress other people. People do things because they consider they have a moral obligation to do them. Etc.

The idea of "learning for its own sake" may be a puzzle even for a steelmanned version of your dichotomy -- e.g., you'd think people do things either because they directly enjoy them or because they are a means to some directly enjoyable end (though I think even that is too narrow, actually) -- but as it stands there are just waaaaay too many obvious exceptions to your dichotomy for the fact that something doesn't fit it to be puzzling at all.

For what it's worth, I think a leading reason for the particular phenomenon of learning-for-its-own-sake is simply that these people have spent years in an education system that consistently praises and rewards them for learning. It would be hard for them not to emerge with a feeling that learning is good for its own sake. (Which it might be -- or it might be consistently valuable in ways that would be hard to predict in advance -- and those may be among the reasons why the education system is that way.)

Learning is often simply compulsive, addicted behavior.

Your brain gets a little kick when the new idea "clicks" in your brain. Oooh, it feels good. You want more. Moreover, learning is an addiction that is generally approved of, so you shoot up with a clear, and possibly self righteous conscience. Oooh, that's the good stuff. Another little treasure of learning random shit you'll never use is that you'll never put your learning to a test. You can get your rush while completely mistaking the point.

It seems to me that you are mixing up two questions. One question is why your mentees study things they think are neither interesting nor important. The other question is why people might be encouraging them to do so. For the first question, why don't you start by asking them? I'm not claiming that they'll have perfect introspection; in particular, I don't expect them to admit to 5, but if the reason is 1, I do expect them to articulate that. But I don't expect many people are motivated by 1; that seems more like the answer to why Amy Chua wants her children to do things. And even if the children know that it reflects parental desires, they may not be able to explain the parental thinking.

Also, to the extent that your goal is to explain the survey, in particular the frustration, I think you can rule out 2 and 4 and maybe 3. Those of 2 and 4 would just move on to another subject.

Another way to put the two questions: why do people do this; and why should people do this?

It seems to be commonly thought that there is a notion of "well-roundedness" which is important in itself. This seems to be a very common reason for learning musical instruments, languages, etc. The person may not find the particular skill or subject to be important, but will do so anyway in pursuit of the well-roundedness that they do consider important.

I've heard claims that engaging in activities that are neither interesting nor important has intrinsic value -- it helps build character, makes one grow as a person, or it just might turn out to be important.

If it builds character, makes you grow, or turns out to be important, then it is important, its importance is just surprising.

What people might actually be doing is operating with the knowledge that they are not good at distinguishing important information from unimportant information. Thus, it seems safer to try to learn as much as possible, in the hopes that that will include the important things.

My impression is that some people believe learning helps keeps their brain/mind alive. They may be right.

Habit?