Elder Mary is one of the policy makers of the Brotherhood, defining its course. Though General Maxson disparages the council's ability to make decisions effectively, they still manage to keep it on course.
Elder Mary is the most concerned about the situation, most importantly General Maxson's claims about an army massing to the north. She requires someone who isn't his friend to scout the northern wastes and return.
While none of the four elders have names, this particular elder is distinguished by the change in the format of her title. While the others are referred to as Brotherhood elders, this particular elder is named "Elder of the Brotherhood."