Boston is notorious for being one of the most expensive cities in the country. I grew up there and those who did not think it was cool would always ask, "But what's there? It's so cold and expensive." Fair enough, as much as I would love to go back, I do believe it would kill my creativity.
Thing is, even if I was not as biased as I am, I would still tell you that Boston is one of the most gorgeous cities in the US. That is especially true in the fall when the leaves on the trees turn colors. That does not even start on how much history there is here and exactly how interesting that makes this city. The good news is, while hotels or lodging may be expensive if you don't know anyone to stay with, there are free things to do in Boston.
Going to Faneuil Hall in itself is free. There are many interesting shops to look at and there are numerous food choices that are very good. Obviously buying something is not free, but just going there and looking around is. As far as historical tours, you can go climb Bunker Hill for free. Be aware that the stairs up to the top is quite a workout. I climb mountains and depending on how good of shape I am in at any given point, I get tired from it. However, once you get up there, the view is spectacular.
Going to the Site of the Boston Massacre and the Boston Common is free. If you are into museums, going to the USS Constitution is free. They do suggest donations; but it is not required. More importantly you will need a valid ID and be willing to pass a security check in order to go.
If you are in town around the 4th of July, their firework display that you can see over the Charles River between Boston and Cambridge is one of the best in the world. Further, if you have kids with you that gives you other options of free things to do in Boston. Obviously most parents would not be that eager to drag their kids up Bunker Hill. So why not pack a lunch (or get a lunch to-go from somewhere) and go sit at the Boston Public Library? Or if eating outside isn't your thing, you can go riding on a bike trail (obviously you could if you didn't have kids and it would not need to be kid friendly in that case...). Also, if you happen to be in town in July, check out Harvard Square for the "Stormin' French Street Festival". This festival celebrates Bastille Day. It features French food, music, dancing and children's events. Finally, also in July, you should attend Artbeat, another art festival that takes place in Somerville. If it can be arranged, July is definitely the best time of year to come to Boston if you are looking to get into the art and culture scene.
For those that are not so art oriented, another fun family event is, on Friday nights, the Museum of Science has an astronomy after hours program on Friday evenings. Well there you have it, what to do in Boston when you are on a budget.