My experience is as a green committee member and co-chair, so I am sure others could identify better questions. Nonetheless, this is a list I would work from as a starting point:
Communication:
Assuming technical competence, ask to see examples of the person's communications to members at prior positions, views on blogs, newsletters, emails, committees; strategy for connecting with members of the green committee and the membership.
Assuming you have a super with technical competence, the key factor differentiating between candidates to my mind is the super's ability to connect with the membership, communicate about problems on the course, navigate problems with committees and rogue members and to balance between ferocious and utterly inconsistent wishes of different members that are used to having their way.
Budget:
Find out what sort of budget the candidate operated under in the past. Ask the candidate to describe any budget cuts she faced, how she implemented them and what impact the budget cuts on the quality of course conditions. Ask her to identify the cost drivers associated with maintaining a budget and what she would investigate first in deciding to cut 10% for a maintenence budget. Determine how capital expenses (such as machinery) was handled at the last job and have her describe the best method for budgeting for such items in the future. Have her indicate whether she tracks labor expenses by task and how she does so.
Get a feel for whether she is a fan of the latest and greatest in equipment, chemicals, irrigation and how she would do a cost benefit analysis whenm deciding whether an expenditure is worthwhile. Get a sense of her familiarity and views on more organic approaches to course maintenence. Have her look at the budget for your course and ask for her analysis of what more information she would want, what she would do with a 10% cut and what she would do with a 10% increase.
Ask her in what circumstances she would seek a midyear increase in budget and get a sense of whether she has done so in the past and how it went.
Conditions:
Ask about maintenence standards and get a sense of whether she has experience setting and managing to written maintenence standards. Ask whether or not she thinks they are a good idea and why.
Ask about her view on keeping conditions consistent from day to day as opposed to priming the course for big events.
Ask about philosophy on pin posiions and maintenence of the practice range and short game areas.
Ask if she has been in a tree removal effort and, if so, how much was handled in house v. out of house. Ask if she has a view as to whether your course needs to remove trees and if so, how she would manage member reactions to such an effort.
Ask her views on irrigation levels, rough height, bunkers, no mow areas, green speeds, use of rolling v. mowing of greens, methods for putting the course to bed for the winter.
Staff:
Ask whether she would want to use existing staff or would be looking to hire a new crew.