Cheap food in Boston???
Where!? Al's South Street/State street Cafes have huge sandwiches for short money. Great quality stuff too. The Small is 10" and is under $6. The large is Obsurd and is $7.50. Sam La Grassas is the best sandwich place in town tho. Get the Rumanian Pastrami... trust me. But those are like $10 ea.
If you are from Texas you won't like the BBQ here, so don't bother. The pizza is the best in the North End. You can get there by foot from Haymarket.
Funny you chose the Omni Parker House if you are on a budget. That place is BALLER. If you get time the Beantown Pub around the corner is a fun night spot w/pool tables etc.
If you want a burrito that will rival the Mexican food available in Texas, go to Andale on Summer st. Order the Chicken Mole or the Carnitas. They have the best Carnitas in the state. looking at about $6-8 per burrito depending on wether or not you get Sour cream and Guac.