A furnished room could be more difficult to find and you may have to buy some small amount of furniture and sell it when you leave Boston.
Beacon Hill might be an option. The northern part of Beacon Hill near MGH is also the most affordable, but the rooms there tend to be very small. You should check craigslist for rooms and shares, or sublets and temporary housing:
boston rooms & shares classifieds - craigslist
boston sublets & temporary classifieds - craigslist
The South End is a possibility but not as convenient to MGH. You'd have to take the Silver line, the Orange line or the bus to downtown Boston and change to the red line. Both of those areas may have apartment shares within your preferred range, but might also tend to be a bit more expensive.
You might have more luck looking in Central, Inman, Porter, and Davis Squares in Cambridge and Somerville. The areas all have a lot of young people and good nightlife and are more affordable than Beacon Hill and the South End. They are also all on the Red Line, which goes directly to MGH and is generally very fast and convenient. Just type "Central" or "Davis," etc., into the craigslist box with your maximum price of $800 and see what comes up. Though not in the city of Boston, they are as much a part of the city atmosphere as neighborhoods in Boston. It is not like living in the suburbs.
BTW, there are undergrads partying in Brighton, but there are more next door in Allston. There are plenty of parts of Brighton that are not noisy or too populated by undergrads. But I don't recommend that you consider living there as it is not conveniently located to get to MGH.
I don't know about the tennis, but there's tennis around.
Good luck!