Boards approve senior hires late in the process, usually when the CEO presents a preferred candidate. By then the useful questions are hard to ask without appearing to undermine the CEO. So ask them at the start, when the search is being scoped.
Has the role been written for the next three years?
Or is it last year's job description with the salary updated? If the plan has changed since the incumbent was appointed, the role should have changed too.
Who inside was considered, and how?
"Nobody is ready" is an opinion. Ask whether anyone was assessed against the role, and against the external market. If not, the Board is approving an external search without knowing whether it needed one.
How was the shortlist built?
How many people were mapped, how many approached, how many assessed, and by whom. A shortlist of three from a market of twelve is a different thing from a shortlist of three from a market of a hundred and twenty. Ask to see the market map.
What does the package assume?
Was it benchmarked, and against whom? A package set from a survey median will either overpay or lose the candidate at offer. Ask for the comparators.
What happens if it goes wrong?
What are the search firm's terms if the appointment does not last? What is the notice period? Who owns the first ninety days?
What did the search firm tell you that you did not want to hear?
This is the best question on the list. A firm that has told the CEO nothing uncomfortable has either done no research or is not being straight. The honest answers are things like: the package is £30,000 light, the reporting line puts good candidates off, or the internal candidate is stronger than the shortlist.
If the CEO cannot answer that question, ask for the firm to present to the nominations committee directly.