Information provided by Bob Ellis of Brookfield, Connecticut indicates that her name was Molly Sherwood.
In August, 1825, the aged grandmother [of P. T. Barnum] met with an accident in stepping on the point of a rusty nail, which shortly afterwards resulted in her death. She was a woman of great piety, and before she died sent for each of her grandchildren—to whom she was devoted—and besought them to lead a Christian life. Barnum was so deeply impressed by that death-bed scene that through his whole life neither the recollection of it, nor of the dying woman's words, ever left him.
From "Struggles and Triumphs or, Forty Years' Recollections of P. T. Barnum, written by himself." The Courier Co.: Buffalo, NY, 1882. "In the month of August, 1825, my maternal grandmother met with an accident in stepping on the point of a rusty nail, and, although the matter was at first considered trivial, it resulted in her death. [N.B., apparently from tetanus]. Alarming symptoms soon made her sensible that she was on her death-bed; and while she was in full possession of her faculties, the day before she died she sent for her grandchildren to take final leave of them. I shall never forget the sensations I experienced when she took me by the hand and besought me to lead a religious life, and especially to remember that I could in no way so effectually prove my love to God as by loving all my fellow-beings. The impressions of that death-bed scene have ever been among my most vivid recollections, and I trust they have proved in some degree salutary. A more exemplary woman, or a more sincere Christian than my grandmother, I have never known."