Hotels in downtown Boston are not cheap and Back Bay is never cheap.
There could be specific factors for those dates such as this is normally when baseball playoffs are of which the Red Sox are usually in the hunt, but really not sure. The Head of the Charles crew races would normally be the weekend after.
Best bet is to list some of the qualities that are wanted and we might be able to give some suggestions. The subway is fairly easy and expansive so getting around should be easy. If price is a big deciding factor, might be best to broaden view making sure it is close to the "T".
Thanks.