I just wrapped up the Air Force 2L internship, so I thought I would share some advice based on my experience (both from things that went well and things I wish I had planned better).
Base selection - I didn't put much thought into this, because my main concern was logistics and cost. I wanted to be somewhat close to the school I attend, because I am involved in a lot of activities that might benefit from me being present at times during the summer. I was able to choose a base that fit this need, and that is great. I even turned down the opportunity to spend the summer in a muuuuuuuch better location, but, as it turns out, that may not have been a good idea even besides being farther away. Bases vary a lot based on their size, composition (e.g., civilian vs. military employees), and mission(s). At some bases, I'm sure an intern could spend the entire summer doing Military Justice if they wished. The base I was at was composed of significantly more civilians than servicemembers, so Military Justice wasn't as busy, but there were many other areas that were busy and/or somewhat unique to the base. If you want exposure to different areas that you might work in as a JAG (and, perhaps, more importantly, would like to be able to speak to your experiences in each of these areas come interview/application time), which base you select/are selected for will have an impact on your options. My suggestion would be to find a current/former JAG and ask them about the differences in the bases you're looking at and/or ask them which bases might fit what you would like to do.
Letters of recommendation - It's difficult, but it would be helpful to identify early on during your internship who, at the base, you would like to try to get a letter of recommendation from. One fact about military life is that people are always on the move - even the civilians. Make sure that the person/people you would like to like you are going to be there during the time that you think you will be able to work with them.
Civilian attorneys - Do not overlook them. We all get excited to work with the JAG officers that have the job we want, but consider that many civilian attorneys working for the office you're interning with could be former and/or retired JAG officers. Also, at least at my base and most of the bases I looked at the rosters for, a lot of JAG officers are very early in their careers, so you might not have a plethora of colonels to choose from to work with. I certainly have little insight into the application process greater than the knowledge presented in this thread, but a retired colonel who has also spent 10-20 years working as a civilian attorney for the branch you're applying to might be able to say quite a lot about your potential as an officer.
Be flexible, especially at the start - Count on not getting a computer for possibly 2 weeks. Or the whole summer. It depends on where you're working within the office, the availability of equipment, how long you're in each section, etc.