Mrs. Tilley plays a small but important role in "A White Heron." Although she doesn't directly impact the plot, her character gives us significant insight into Sylvia's family life and background.
First off, we learn how Sylvia ended up in the New England countryside with her grandmother. Although Sylvia was never happy in the city, she wasn't the one who decided to move to the country—in fact, it was Mrs. Tilley who "had made the unlikely choice of Sylvia" (1.3) out of all of her grandchildren. This tells us that Mrs. Tilley is both a matriarch for her family and quite perceptive at that. Not only did someone send her their kid, but she picked the right kid to have uprooted and sent her way.
Mrs. Tilley's family stories fill in a few more blanks. We learn that her other children—Sylvia's mother included—are now scattered across the country is vastly different circumstances. Most notably, there's Dan, who was once as in-tune with nature as Sylvia and is now a "great wand'rer" (1.15). Although Mrs. Tilley never goes into detail about what that exactly means, we get the sense that Dan's life is a bit out of control.
Given the similarities between Uncle Dan and Sylvia, would it be that far-fetched to imagine Mrs. Tilley seeing a similar fate for Sylvia? Does she realize that Sylvia's sensitive nature makes her an easy target for manipulative men, as we see with the hunter? Despite not receiving concrete answers to these questions, we get the sense that Mrs. Tilley knows a lot more about her granddaughter than it seems at first glance. After all, she's the one who picked her for the country life.