This kind of article is marvelous. Dr. Graham is (ostensibly) advising a woman, soon to be a mother. It assumes the Bible is a model for behavior--that there's no gap between one ancient text and another, and between any biblical text and our own time. I understand the Bible to be authoritative but I don't expect that it will tell me how to behave.
So what does the Bible say about motherhood? Two quotes are given: "Honor your father and your mother, as the Lord your God has commanded you" (Deuteronomy 5:16). And, "Listen, my son, to your father's instruction and do not forsake your mother's teaching" (Proverbs 1:8).
Take them at face value. What do they say to you? Both mention the parenting of father and mother. One is a command, the other a piece of advice. God's command to honor father and mother is non-negotiable; the other passes on something that will help you get along as you grow up. Jesus would have known both passages and would have kept the 10 Commandments. As we know from the gospels, he discusses application of this command in Matthew's gospel. What is being discussed are the ways to honor parents. If you take both passages together, the latter is the pragmatic benefits of the former.
Nothing of this is evident in Dr. Graham's arguments: The first text undergirds the notion that "motherhood is one of God's great gifts to the human race." And the second supports the assertion that "mothers especially are equipped to comfort and teach us."
To me, this is a great example of making the text say what one wants it to say (in this case about mothers) rather than taking the text at face value. Then the larger question is, does the Bible actually say anything specific about mothers? Does it in fact ascribe to mothers at any time and place particular roles?