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.

1. 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)
2. Benchmark the product against the competition. This will help you tremendously to get a strong foothold of the industry.
3. 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.
4. Participate as the honorary QA tester. You will know the product inside-out. You will also know the process, the politics and bureaucracy involved.