The US embassy say that if you have a British passport and are travelling on the visa waiver then it only needs to be valid for the duration of your stay.
This is what it says on the website
Passport Validity
If traveling under the Visa Waiver Program, the passport must be valid for at least 90 days from the date on which the holder enters the United States. If not, the holder will be admitted until the expiration date on the passport. Note: Passport holders from Andorra, Brunei, and San Marino must be in possession of a passport valid for at least six months from the date of departure from the United States, even if traveling visa free under the Visa Waiver Program