Try a little bit of everything.  Do what you can to include yourself in meetings, conference calls and email chains and immerse yourself in the case or deal.  Focus on taking in what everyone is doing--the partners and the associates, but also the clients and opposing counsel.  Try to think about what you find attractive and unattractive about the different practice areas and the different roles everyone is playing and use that to inform your career choices within and outside of the firm.

Chances are, you're going to have plenty of downtime and not be terribly swamped -- I recommend taking that time to learn the boring, nuts & bolts procedures. What I mean is: how internal filing works, how your corporate library works, how your word processing/ resource department works, how to make a closing set, how to pull a closing binder, how to search for documents, how to use your firm's document templates, etc. When you actually start as a first year, I guarantee that you'll be thankful you figured that stuff out as a summer when it's 10 p.m. and the senior associate you're working with asked you to pull precedent from the corporate library or it's a Saturday night and you're trying to figure out how to get a markup couriered to the partner and the office is nearly empty.

Work hard, but maybe not in the traditional Biglaw sense.

(1) Work hard to get to know your peers, as they'll be your biggest resource at the firm. And you won't have another 12 weeks like this in a very long time.

(2) Work hard to get to know the practice groups and the lawyers working in those groups, as you may have to make a decision as to where to practice (or at least you'll want to know who you like working with/don't like working with).

(3) Work hard on the matters/cases/files you're assigned/you procure... but don't take on too much work, because (1) and (2) above are more important at this time.

Try EVERYTHING. This is one of the last times when you can try any type of matter you want so take advantage of the flexibility! Also, agree with getting to know your fellow summers - so helpful when you have a random question on something in the future and don't want to spend all night researching.

Take advantage of meeting the first and second year associates that go out with the summers for events/drinks. It's a big help to have friendships with the third/fourth year associates upon returning to the law firm. 

Separately, if your firm offers rotations through other offices, the summer is a great time to get a feel for the differences between the regional offices of a firm. 

Just know that people's memories are long at law firms, especially if they are bad memories.  You don't get to press a reset button when you start as a first year.  So make sure you don't do or say anything that will leave a bad impression of you.  Not only will your fellow summer associates remember, but so will partners and associates you interact with.  It sounds like very common sense advice, but you'd be surprised by how many people screw up (esp. at drinking events) and what doesn't help is that news often travels fast in a firm.  

To echo some of what's been said by others, I think the most important thing to do during your summer is to try a variety of projects and practice areas in order to get a sense of where you'd like to start upon your return.  That is the most immediate and pressing objective, assuming you haven't already decided what sort of law you'd like to practice.  

If you're already set on a practice area, then you should focus on getting some real work on matters and with teams that you like.  Litigation matters at large law firms tend to last for years, so if you get put on an interesting case as a summer, chances are it will still be around in a year or so when you return to the firm.  If you do good work as a summer, the case team will probably welcome you back with open arms.  The opposite of that is also true.  So while you should not kill yourself during your summer, you absolutely should dig in on a few matters, do good work, and start to establish a reputation as someone who can be counted on.