Eight years back, I answered this question on Quora and some have found the answer useful even today. Let me just reproduce the answer here:
A summer internship can last for a short duration like 10 weeks. The problem is 10 weeks is a very short time, but not too short to make an impact. Hence focus on the tactical areas of product management and very less on the strategic areas.
- You must try to do some data crunching – qualitative and quantitative and provide your recommendation on which features monetize better over the others. This helps the PM to be armed with factual reasons as to why a feature should or should not be built (invested)
- Benchmark the product against the competition. This will help you tremendously to get a strong foothold of the industry.
- Ask for the opportunity to prioritize bugs. This in some places is a dirty job because of the magnitude. This will tell you health of the software and also when you objectively prioritize you earn the respect of Engineering and QA – the value of which is worth its weight in gold.
- Participate as the honorary QA tester. You will know the product inside-out. You will also know the process, the politics and bureaucracy involved.