do you have any idea/s?
a : Why is he so tired? Do you have any idea?//Do you have any ideas?
b : Because he worked hard all day today.

Question

Answer

''Why is he so tired? Do you have any idea?'' sounds better to me.

Answer

I would use "do you have any idea" when asking the question all in one go.

E.g. Do you have any idea why he's so tired?

But if I were to break it down into two questions, I would favour "any ideas."

E.g. Why's he so tired? (Do you have) Any ideas?