A partner interview means that you are almost in, but not quite. Deloitte Consuting is very concerned about "fit". That is what the partner interview is usually about. Rather than giving you a case or trying to grill you on your resume, it will most probably just be a relaxed conversation about your interests outside of work/study. They are trying to get a feel for you as a person, and assess whether you will gel with the collegial culture.
My advice to you is simply relax and enjoy yourself. My interview veered off into strange directions like looking at the performances of sporting teams I supported and which cars I thought were really cool. Again this had to do my interests, so don't worry if you're not a petrolhead!
The partner did ask me about what I considered my greatest achievement. If possible try to stay clear of academic scholarships and the like. Use something that demonstrates a rounded personality.
That's about it. Be aware that this was my experience. Your partner may decide to try a completely different approach. If HR isn't sick of you yet, you could try to ask them for details about the type of interview. Good luck