A common characteristic of fixed phrases and idioms is that their grammar is not so obvious, but here, it should be clear that 'of' is optionally omitted: I have no idea (of) when/where/what/how/why he is doing it. He left no clue (about) when/where/what/how/why he is doing it. It is no secret (no prep) when/where/what/how/why he is doing it.
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.