Ideas for project pitch?

I'm a rising senior off from school this year, back home in a city of ~1 million. I'm aiming to fill the last three months of the spring with some local finance experience. I plan to call and visit some small (1-10 person) VC firms as well as a bunch of asset management firms and essentially offer to work for min$/unpaid so that I can get some experience.

For my pitch I want to show them an example of a sample project, i.e. basically show them that I can actually be useful and help them out. My question for you is: what exactly do you think I should include in that project pitch? An example of valuation or other reporting/research on a company? What's what should I bring to the VC firms as opposed to the AM firms?

To put the question more accurately, maybe, what specific tasks could I offer to a small local VC/AM that both (1) would be relatively basic and routine enough that the FT employees would want to delegate them to me (2) and would still give me exposure and experience that would educate me for the long-term and provide a bullet point or two for the resume (i.e. doing valuation rather than brewing coffee)?