When asking a peer or colleague why they are pursuing a certain research direction, I have often heard the answer "because it's fun". This bothers me: Shouldn't we all be capable of providing an honest narrative explaining why our chosen research direction is of value
What makes you think this is not an honest narrative? Why should I come up with some (possibly fabricated, ex post facto) justification for why I am doing my work, in a casual conversation, if the real answer is that I enjoy it? If I were asking this question I would appreciate an honest answer, and if "because it's fun" is honest, then that is helpful. You could proceed by prodding more into what aspects of this research in particular are enjoyable.
On the other hand, the last part of your statement, explaining why our chosen research direction is of value, is a totally different question. There is still some ambiguity about what this means (of value to the researcher? of value to society?), but I think this is more likely to lead to a discussion of the technical merits of the research, and what are the promising motivating applications.
But shouldn't one have a strong explanation for the value of one's research other than one’s own enjoyment?
It seems unlikely that you can be a successful researcher without having such an explanation; you are, indeed, forced to flesh out such arguments in detail in the introductions to papers and grant proposals. On the other hand, it is possible to not have such a strong explanation, particularly for early researchers (starting PhD) or senior researchers (e.g. already won all the awards in the field, now just pursuing some fun idea for its own sake).
So with respect to the question in your title,
Is “because it's fun” an acceptable justification for choosing a certain research path?
I think it has strange moral overtones; why would it be unacceptable? But I can see that you might want to hear in more detail about why it is fun, and separately from that, why it has value, and I agree these are useful details.