If he is receiving Contributory ESA then the answer is no he will not receive any Mortgage payments. If he is receiving or would be entitled to Income Related ESA then the answer is yes. It will most probaly be after 13 weeks but some claims are still 39 weeks. It is not cut and dried. Mortgage protection Insurance would also be taken into consideration. Also if he has a partner working over 24 hrs this would automatically exclude him from Income Related ESA.
Tell him to call his local Benefit Delivery Centre and the Mortgage Section there would be able to advise him if he is eligible.