Hester is not transformed or renewed in any way by her punishment or this experience aside from her physical appearance.  She loved a man who was not her husband, and despite her public ignominy she loves him still.  Unlike Dimmesdale, it's clear Hester would have committed the same sin/crime of adultery with Arthur if she had been given the chance.  To that extent, then, Hawthorne portrays her as an independent woman, beyond the morality of this Puritan town.  The people change how they see her and interact with her, but she is essentially unchanged from beginning to end. 

"The letter has not done its office" Hawthorne writes near the end of The Scarlet Letter. Hester Prynne, though limited in the position that she can hold in her community, does not conform to any typical role of a woman. For, the fact that her letter is redefined a couple of times is indication that there has been no categorization of Hester.  Her independence is symbolized in little Pearl, whose passionate nature is reflected in the reds and bright colors that the child wears.  It is Hester's pride that sustains her, not any meekness or a complaisant nature. Hester is no Puritan and her respect for the Puritan code has never overcome her independent passions.  She tells Dimmesdale in Chapter XVII,

"in all things else, I have striven to be true!  Truth was the one virtue which I might have held fast, and did hold fast, through all extremity....A lie is never good, even though death threaten on the other side.!"

Having learned from her sin, Hester becomes strong:

The scarlet letter was her passport into regions where other women dared not tread.  Shame, Despair, Solitude!  These had been her teachers,--stern and wild ones,--and they had made her strong...."

From her experiences, Hester emerges as an independent individual, capable of helping others, but respected by them.  She takes initiative in trying to save Dimmesdale from destruction, playing the role of the stronger, not the meek.  Hester emerges as a much more modern woman than that of Colonial times; she survives repression and hardship and takes her own place in her  community.

When we consider this novel from a feminist view point the crucial question we need to consider is whether Nathaniel Hawthorne has portrayed Hester as an admirable figure, who has transcended the limits placed upon women and their experience in the mid-nineteenth century, or whether she is ultimately to be seen as a sinful transgressor, who has deserved to be made an outcast. Some feminist critics might use this issue in order to measure Nathaniel Hawthorne's response to contemporary agitation to the social advancement of women.

Either way, it is clear that Hawthorne's direct and unsympathetic portrayal of the Puritan hierarchy of seventeenth-century Boston is something to be applauded by feminists. I think we can safely say that Hawthorne clearly disapproved of the sternness and inflexibility of New England Puritanism, but arguably he didn't go far enough for many feminists in that he wanted change but only in a fashion that would leave the central relationships between men and women unchanged and patriarchal society intact.

Interestingly some critics have argued that Hawthorne's characterisation of Hester actually leads to the promotion of female stereotypes. Some argue that he has depicted Hester as a temptress, a kind of Eve luring Adam to his downfall. Hester is alternatively presented as a kind of Florence Nightingale figure, tending to the sick and the needy. This equally endorses a well-established stereotype of specifically female roles - the woman as the wife and the mother, the nurturer, the carer, where her role is contained narrowly within the domestic sphere.

So when we think about what Hawthorne actually wanted women's role to be, I think we can safely say that he condemns the Puritan world of his novel, but at the same time, he did not necessarily want to overthrow a patriarchal system that gave men the power in this society.

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.