Engaging the interviewer. How?

Good question. It's actually really hard to give general advice on. Again, a bit like going on a date - how can I make sure he/she likes me?

All social rules and norms apply -- what kind of candidate would you like if you were the interviewer? Seriously: Not being socially awkward is already a plus ;)

And you ask -- "How can I ensure that the interviewer is having fun (...)?" Easiest answer to this is, did you have fun? If yes, chances are high the interviewer had, too. In all of my best interviews I had a laugh together with the candidate.

Few concrete tips:

• Connect over something. In most firms, you get a short bio about the interviewer prior to the interviews. Read it, and remember just 1 thing you two have in common. (Same university? Playing tennis? Same home town? whatever it is) Bring it up at the right time. The interviewer will try doing the same.
• Listen. I hate to even write it here because you will have heard it so many times already, but an interview is such a high-stress situation, most people forget. If the candidate is not listening, there is zero engagement from my side as it's pointless. As an example, if your interviewer says something about your case structure or any feedback at all means she wants to help you. Tough luck if you don't notice. Answering the damn question is the same category. I ask "name 3 points about ...", and get 2 or 5 points back from the candidate. "Give me 1 sentence", I get a 2 min monologue back. Or 1 sentence and a long explanation. All the time :)
• Name the elephant in the room. Less obvious, but extremely powerful. You know your weaknesses, and chances are your interviewer will find out about them after 5 mins (if it takes that long). Addressing them directly and honestly is a power move and makes you super likeable. Example: I once had a candidate who seemed extremely shy, to the point where I was even unsure how to do a full interview with her. Pretty early on, I asked some question about her strengths, she said "attention to details, working under extreme stress, and being underestimated all the time because I seem so shy". Another time, I had a super arrogant guy who I immediately disliked (there are these people). Asking about something related to weaknesses, he mentioned that he is quite insecure, and while trying overcome it, people now see him as arrogant. As an interviewer I cannot dislike someone who just "confessed" and is committed to working on these issues (game over without the latter part). Another word of caution: Faking it will backfire ;)
• Ask questions. At the end of every interview, there is a chance to ask questions. I can't think of anyone we ever hired who had no questions. Ask questions that make the interviewer look good. Ask interesting questions to stand out. Examples I personally dislike: "How does this international exchange program work that your firm offers?" (frankly, I never really knew about all of these programs, they change constantly, so your question makes me look bad). Also, I didn't really like the work-life-balance questions I got all the time. Not because I think it's not an issue, but my opinion is that you should figure that out before you apply. Good questions I remember were "If you would be CEO of BCG, what would you do different?" and "If you weren't a consultant, what would you be doing if money was no issue"? But again, this is highly subjective; other interviewers may hate them. "What was your most exciting project" is overused, but a safe bet.

Hope this helps - good luck with your interviews!