From the British Foreign and Commonwealth Office.

<<<< Does my British passport have to be valid for six months beyond my date of departure from the United States?

No, if your passport is not valid for at least six months beyond your date of departure from the United States, it will not affect your eligibility to travel. The United States has an agreement with the United Kingdom automatically extending the validity of a passport for six months past the passport's expiration date. Therefore, your passport need remain valid only for the duration of your stay in the United States. >>>>

If you choose to renew earlier REMEMBER you need to renew your ESTA also.