Given the time and the economic context, I assume Mary refers to the couple's nanny/wet nurse. I suppose one could build other allusions—to a ghost on the property, etc.—but I really think this one is just literal. (It is possible that the narrator is already crazy, and Mary's there to take care of her too.)
Mary may well be an ironic allusion to the Virgin Mary. The narrator remarks, "It is fortunate Mary is so good with the baby. Such a dear baby!"
This musing may seem innocent, but it underlies the unnamed narrator's uneasy feeling about being a mother. Far from the angelic, blissful state society expects mothers to experience upon the birth of their children, the narrator finds motherhood suffocating and isolating. She gives the pat response to expected from any mother, but her increasingly declining mental state says otherwise.