Can this be used in the rather idiomatic way it is used in English, or is it just saying that the speaker does not have an idea to offer for, say, a solution to something?
It means the same as saying "I dunno" to someone asking a question. Is this what you mean by idiomatic way?
You can also say: "Weet ik niet" or "Weet ik veel", the latter being more rude/unformal, to mean "I dunno" :) EDIT: Both are unformal actually hahaha, but the 2nd is more rude
Weet ik niet is neutral. But indeed the second one it is quite informal, maybe a bit rude, depending on the context and for sure people won't be happy with this answer, it roughly is the Dutch equivalent of How would I know? or I haven't got the faintest clue.