When you are asked "what is the most difficult project? and how did you overcome the difficulties", what is the expected answer for this?
When you are asked:
What is the most difficult project? How did you overcome the difficulties
They are really asking:
What is your process in dealing with challenges/issues
And, typically, what they are looking for is someone who can isolate, identify, and then solve problems. Ultimately, it doesn't matter how big of a difficulty you had with any particular project, but the process of how you overcame that difficulty.
Also, it does matter what kind of challenge it was - if you're looking at a team leader or manager role, it might be better to talk about a people issue rather than technical - "working on x, and one of our developers wasn't quite meshing with the rest of the team and it put our delivery at risk. So I took him aside and offered to mentor him, which he accepted, and within a week his productivity improved and the team delivered on time and we all got a bonus."
If you're looking at a developer or architect role, then highlight something more technical.
The keys would be how well can you give the context of what was the project, what made it hard and what did you learn from it among other things.
The interviewer is seeing whether or not you'd speak negatively about the project or a past employer, how well do you know your weaknesses, what have you done about them and how would you handle this situation should it happen again in the future. There are more than a few mines you could hit here that lower your chances of getting the job.
What would be a good answer to this is going to vary greatly depending on the seniority of the postion you are talking about and the needs of the particular position for that particular company. But in general they are looking to see how you deal with challenges and what your technical abilities really are. So the expectation is going to vary based on your own experience level or claimed experience. For instance, suppose I see a resume that claims lots of different knowledge and says all sorts of exciting things the person did, but in answer to this question, he describes how he once had to create an Access form that used a data lookup. In this case, his claimed experience is out of synch with what he said was actually a difficult project. You would be amazed at the number of people who claim to be really experienced but who can't describe anything challenging they have actually done. This question helps weed out the resume liars.
If it is an entry level postion talking about your most challenging class project or some work you did as an intern or even a personal project realting to your profession would be fine. Talk about what made it challenging, how you solved the problems of the project and what you learned from it.
After entry level, the expectation would be to discuss a work-related problem. If you have 3 years of work experience I don't want to hear about how hard your C# class was. If you are looking for a journeyman level, I would try to describe a project that shows you are ready to move up from entry level.
If the job is senior level, I would expect the challenge to be more than something technically difficult, but something technically difficult that had some elements of soft skills as well such as project management, dealing with difficult people, pushing back requirements that were inadequate, etc. At the senior level, technical skills are only 30-40% of the job in my experience.