This is one of the most important parts of the interview.
It's a chance to demonstrate what you know about the firm, truly understand the office/culture/fit (remember this interview is two-ways!), and also build some rapport with the interviewer leaving on a high.
Good interviewers use this to test candidates EQ (I certainly did, and poor candidates either asked irrelevant questions or didn't have any in the first place)
Certainly do not waste the opportunity by asking silly administrative questions or HR-type questions.
Ask insightful questions related to topics:
- Interviewer - why he/she joined that firm, how their experience has been, what they are working on, what would they change about the firm if they could (consultants love talking about themselves so endow them with the opportunity!)
- Industry - how they have seen the industry change, what the state of affairs is for consulting work, what are the big opportunities in the near future
- Firm - how staffing works, what sectors are known for what kind of work (some sectors do more ops vs. strategy work for example), who the firm competes with, what differentiates it from others
- Role - what to expect in first 6 months, what to expect on first engagement, any advice