For $1200, you won't be able to live on your own in biking distance of MIT. You might be able to find a crappy studio for $1600. $1800, you might find a more acceptable studio or a 1-bedroom. At $1200, your best hope is a room in a 2 bed, 2 bath...

Other notes: most of Somerville and Cambridge are bikeable to MIT or close to the red line. Anything close to the red line, you can take the train easily. Any other T line will get you there, but it'll take significantly longer since you'll have to transfer, or Lechemere on the green line will be maybe a 20 minute walk. Allston has a lot of apartment buildings, and is bikeable to MIT, and you can find some places there on the cheaper side.