What is Nathaniel Hawthorne's view on woman's role in the family and in society in The Scarlet Letter?
I think it can be said that Hawthorne, for his time, took a mostly pro-feminism stance in The Scarlet Letter, as he made his lead character a female who not only survives, but ultimately thrives in, a society in which she socially does not belong.
Consider first, what the Puritan society viewed as the woman's role. When you look at evidence from history and details in the story itself you can see that the Puritans valued several things for their women: adherence to Biblical rules/boundaries; married to one man; child-bearers and child-rearers; mild mannered and subservient to men in public (which created critical gossip w/other women in whispers); housework and feminine hobbies were acceptable, even for pay, but women did not do hard labor (note also that Hester Prynne was never called upon to embroider/sew anything having to do with a wedding). Women were not active in government, could not be religious leaders, and did not have an identity outside their husbands (consider Hester Prynne's original marriage to Chillingworth).
Then, when you look at Hester Prynne, you can understand how Hawthorne wanted to paint a picture of a heroine who did not conform to societal standards. First, she was a single mother who was successful at raising her child. She attempts to give Pearl proper "catechism" but ultimately raises a child who can think independently and stand up for herself. She was the most successful/sought out seamstress in her colony, and was able to make an actual living off of her skill. Therefore, she did not need a husband, either for her livelihood nor for a father to her child. She ornately decorated her scarlet letter, and bore it until the day she died. It is as if Hawthorne wishes to present Hester as a woman who deeply desires to do things the "right way" (she accepts that she has sinned and deserves the punishment) but sees the hypocrisy in society and wishes to rise above it.
Looking at the above comparison, I urge you to form your own synthesis and conclusions of Hawthorne's view of women, but I would certainly argue that he was less traditional than the society in which he was raised and was in favor of women using their intelligence in any aspect of society (the family and their role in the community) to help shape their own personal identity and benefit others. In a period in history when women were not considered equals to men, Hawthorne was very revolutionary and progressive. While he may have agreed that a woman could be most successful "in the home" and raising children, he certainly did not believe that was all she could do, nor that she does it because it was an easy task.