re: Boston next Fall-- Any budget friendly hotels?

As others have mentioned, have you checked AirBnB? Usually can find options cheaper than hotels in the Back Bay/Beacon Hill area.

If you’re willing to stay outside the city but with access to a T stop to ride into downtown, check out Quincy. It should be a little cheaper.

Edit: if you want to give a ballpark budget I’m happy to help out in your search.