Biddy is quite simply just a good person. She is gentle, loving, soft-spoken, wise, and efficient. She tutors Pip and does a fine job with him.
Later in the book, Pip realizes just how great Biddy is. Previously he had been somewhat blind to her character as he was blinded by class distinctions.
One reason Biddy is a good nurse is because she is a loving, trusting soul of discernment. When Biddy was released from Mr. Wopsle's great-aunt's service (because the great-aunt had overcome the "bad habit of living"), she went with "a small speckled box containing the whole of her worldly effects" to be a comfort to Joe and Pip in Joe's household. Her first "triumph" of service to Pip's invalid sister was an act of discernment and intuition and was to interpret the meaning of the T-like hammer (or hammer-like T) Pip's sister persisted in drawing on Pip's chalk slate. Biddy correctly interpreted the symbol as representative of Orlick (the blacksmith); thence began the processionals of "Orlick's slouching in and standing doggedly before her." Biddy's loving and trusting nature is revealed when, in a callous moment, Pip forgets her role in his early education and she, with a teardrop on her cheek, reminds him of it, saying: "I was your first teacher though; wasn't I?" Her tenderly caring qualities are shown in how attentive she always is to Pip's sister:
"Ah, poor thing!" replied Biddy. It was like her self-forgetfulness to transfer the remark to my sister, and to get up and be busy about her, making her more comfortable....
In Biddy's "unimproved" days, Pip was most aware of her "extremities": "her hair always wanted brushing, her hands always wanted washing, and her shoes always wanted mending and pulling up at heel." Pip notes that she was an orphan like he was and living with her distant relation, Mr. Wopsle, and that she maintained for Wopsle's great-aunt "a little general shop" that was, before Biddy's arrival, in sad disarray. The only shop order was to be found in a "greasy memorandum-book kept in a drawer, which served as a Catalogue of Prices." With this "oracle," Biddy established order in the shop shelves, prices, goods and transactions. Biddy became a good housekeeper, then, by the combination of her dedicated labors for Wopsle's great aunt with her natural propensities for astuteness, orderliness, activity and industriousness.
Biddy's greatest qualities are her intelligence and her empathy. She is someone who always cares for others, while deprecating her own troubles. In many ways, Biddy would have made a more deserving hero for the book than Pip, who tends toward self-pity, at least after he begins to visit at Miss Haversham's. In Chapter 17, Pip tells Biddy about his plan to become a gentleman and of his love for Estella:
“We talked a good deal as we walked, and all that Biddy said seemed right. Biddy was never insulting, or capricious, or Biddy to-day and somebody else to-morrow; she would have derived only pain, and no pleasure, from giving me pain; she would far rather have wounded her own breast than mine. How could it be, then, that I did not like her much the better of the two?”
Biddy, of course, knows Pip's heart better than he does himself (and everything else: earlier in the chapter Pip remarks, "In short, whatever I knew, Biddy knew") and is characteristically understanding:
“Biddy," said I, when we were walking homeward, "I wish you could put me right."
"I wish I could!" said Biddy.
"If I could only get myself to fall in love with you,—you don't mind my speaking so openly to such an old acquaintance?"
"Oh dear, not at all!" said Biddy. "Don't mind me."
"If I could only get myself to do it, that would be the thing for me."
"But you never will, you see," said Biddy.
Biddy, who is smarter than Pip both intellectually and emotionally, is prevented by her insight from hoping that Pip might love her. She is a rare mixture of emotional acuity and rational clear-sightedness. The qualities that make her ill-suited (or too good?) for Pip make her an excellent nurse and confidant.
Biddy is a good nurse and housekeeper because she places the needs of others before her own. She is naturally a loving, compassionate person and has a maternal instinct to take care of others. While she may not be book-smart, she is a highly intelligent and sensitive human being who anticipates and fulfills the needs of those around her.