I reviewed the FY2010 schedule and found that there are two 11 week classes scheduled. One begins during the last week of Jan and the other begins duing the 2nd week of October (website:
http://www.jag.navy.mil/njs_curriculum.htm The curriculum link (right side) has the schedule.
I'm guessing that he will finish TBS sometime in September. He should have his orders sometime early September. You can use those orders to request base housing (if it is available to student officers). Go to their website. There will be contact info so that you can inquire about housing when you arrive. Since you are TBS base housing, I doubt that you will be allowed to stay at Quantico, soooo, it is highly likely that you will be able to go with him.
He may be on Temporary Duty Orders (TDY) for training since the school is 11 weeks long (less than 6 months). This means that he will likely receive per diem for each day that he is there to include temporary quarters (food and lodging pay). There are no restrictions on you traveling with him, but the govt will not pay for your transportation or shipping furniture if he receives this type of orders. The govt should pay for storage of your household effects while he/you are at Rhode Island.
If he receives Permanent Change of Station (PCS) orders, govt pays for shipping of household goods and your transportation.
I and my wife/daughter have traveled to various TDY for training destinations while I was on active duty in the Corps - no problems and per diem was sufficient. Cost for food would be same as if we were at our permanent duty station (from my Marine Corps paycheck). We sort of looked at it as a "fringe" benefit. He will be quite busy, but since he is an officer in a Navy school, the schedule will not be very grueling. There will be homework, so his personal time may be restricted a bit, but not as bad as TBS.
Welcome to the Corps. Semper Fi.
Lieutenant Colonel, US Marine Corps-Retired