Hi, Spidergal! You're in for a treat of a city.
As Jespah said, I like the Morrison House
in Somerville. Somerville is a suburb, maybe a 20-minute subway ride west of Boston. It has lots of students in it, which means good ethnic restaurants around its little center. The Morrison House is a bed&breakfast, five walking minutes from the subway. (I second Jespah on being car-free in Boston itself.) It's a pretty wood building, built around 1900 I would say. It has a friendly owner and a shy cat that will eventually let you pet it because it's also curious. Most of the rooms are upstairs.
At $100-170 a room per day, the Morrison House is cheapish for the Boston area, though not for anywhere else in the nation. If you want something real
cheap, you may want to look into youth hostels. I have no experience with these.