My Mom died intestate at Christmas. My brother and I are the only survivors and he wants to relinqish his share of any inheritance to me. Can he do this at any point - the grant of administration from the Probate office is not yet issued. What does he have to do and what are the implications?

Depends on the state. In general your brother can disclaim his rights to the inheritance, bit it must be done within a certain period after the death of the decadant as specified by state law (typically within 9 months). It must be done in writing, and he must not have had any financial benefit from the assets he is disclaiming. He may not dictate who gets the assets he is dicaliming - that is controlled by the terms of the will or in this case (since your mother died intestate) by state law.