Please heed Eugeanie's advice: "Don't be lured by a false economy."

Would you recommend visiting London and staying at the Hampton at Gatwick Airport, or the DoubleTree at Heathrow?

To experience Boston, splurge, if you can, and stay at Hilton Downtown/Faneuil or Hilton Backbay. Both show availability at 70k next week. Downtown/Faneuil is now showing 56k for 5-nite stay. A lot of points, but you will walk out of hotel and be where you want to be.

For CLOSEST beaches, see attached article for closest beaches to Boston:
https://www.boston.com/culture/new-e...ch-near-boston

For BEST beaches, you need to get to Cape Cod. Rent a car and drive to National Seashore (midweek, not weekend, due to traffic). Or, take the ferry from Boston to Provincetown and Race Point Beach and Dunes, with option of continuing further into National Seashore.
For PTown ferry, see:
http://www.provincetownfastferry.com