Learn English – Do you have any thoughts/thought

The correct use changes depending on the sentence:

Do you have any idea how to do this?
Do you have any idea what to do?
Do you have any ideas for me?
Do you have any ideas for how to do this?

It seems that, if the singular or plural noun (idea) is directly connected by a subordinating conjuction (how / what / where / which / that), you use the singular; if it's modified by a prepositional phrase (for...) or by nothing at all, you use the plural.

Without using all that grammar jargon, I'd say use the singular if it's followed by how, what, that, who, whom, whose, which, etc, and use the plural otherwise.

Who can be either an interrogative pronoun ("Who is that?") or a relative pronoun ("The man who sells fruit"). Neither interrogative pronouns (question words) nor relative pronouns (which/that/who and variations) are bound to grammatical number by themselves. The plurality is instead bound to the object in question.

Examples:

"Who is that man?" - singular due to "man"

"Who are those people?" - plural due to "people"

"The man, who is sitting there,..." - again singular due to "man"

"The men, who are sitting there,..." - plural due to "men"