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I think the division into problems and exercises usually seen in mathematics texts would be useful: A task is considered an exercise if it's routine application of previous material, it's a problem if it requires some kind of insight or originality. So far most of the Koans have seemed more like problems than like exercises, but depending on content both may be useful. I might be slightly biased towards this as I greatly enjoy mathematics texts and am used to that style.
"Problem" suggests something different in philosophy than in math. A philosophy "problem" is a seeming dilemma, e.g. Gettier, Newcomb's, or Trolley. So I'd suggest "exercise" here.
"Exercise" dominates "kōan" in that both have the sense of something to stop and think about and try to solve, but ① "exercise" avoids the misconstrual of Zen practice (the purpose of a Zen kōan is not to come up with a solution, nor to set up for an explanation), ② the Orientalism (the dubiosity of saying something in Japanese to make it sound 20% cooler), and ③ the distraction of having to explain what a kōan is to those who don't know the word.
EDIT: The claim that a purpose of a Zen kōan is not to come up with a solution appears to be a matter of disagreement, so discount ①. I think ② and ③ stand, though.
The account in the Wikipedia article says differently:
According to the history of the word given there, it originally meant accounts of legal decisions (and literally, a magistrate's bench). In Chinese Buddhism it came to refer to snippets of dialogue between masters. From there it mutated to the contemplation of mysterious sayings, and eventually to what looks very like an exercise in guessing the teacher's password, with authorised answers that were specifically taught and had to be given to acquire promotion in the Japanese monastery system. (I have this book, which is subtitled "281 Zen Koans with Answers".)
The modern meaning of "koan" dealt with in the section "Koan-practice" describes what looks very like Eliezer's intention in using the word here: a problem that cannot be answered by merely applying known rules to new examples, but requires new thoughts and ideas: a problem that begins by seeming impossible: a problem that cannot be solved without in the process learning something that one has not been taught.
Perhaps there is, somewhere, a better word, but I think "koan" will be hard to beat.