Ok, here's my Boston background first of all: Lived in the City for 5 years and have commuted to/from Boston for work since 1983. I commute from the North but have been to locations South and West of Boston also.
Yes, Braintree has a T stop (red line) but it will be packed at commuter times and still quite a distance from the City.
Driving from either the South (Braintree) or North is NO FUN at commuter times. That's when you'd most likely be on the road. Even with the new roadways leading into the City it's still bad. Yes, it looks like a straight shot in .... bumper to bumper at various speeds though. Bad on the nerves. Then there is the issue of where to park the car in Boston. Meters are scarce and garages are expensive (average is about $20-$25 day without validation).
If you can't find a reasonable price in Boston try for Cambridge, Charlestown, Brookline or Allston/Brighton for closer and safe alternatives. All are close to T access that won't take lots of time. You can even grab a taxi into the City if you wish. Many places offer shuttles to and from Logan also.
As a further alternative there is Woburn. It's North of the City and has both T Commuter Rail and Logan Express Bus transportation with plenty of parking. The Commuter Rail train into North Station takes about 20 minutes. It drops you off at North Station/Fleet Center, within walking distance of the North End, Charlestown and Faneuil Hall areas.
Ok, I did a quick price search for you. I looked over the list of hotels and selected a few below.
Places to Stay - April 9 to 12
* Royal Sonesta Cambridge - right across the river from Boston, near MGH and Science Museum
* Hyatt Regency Cambridge - near river and Brighton
* Boston Omni Parker House - downtown near shopping, Boston Common, T and Faneuil Hall
* Copley Square Hotel - near Prudential Center, shopping, dining, tours and great T access
Crowne Plaza Woburn - near malls, movie theatres, Commuter Rail to Boston, Rts 93 and 128/95 and Logan Bus
AmeriSuites Medford (new)- right off Rt. 93, not much around the hotel though (residential)