Industrialization happened long before the organization of women's rights movements. As per the previous answer, before the industrial revolution, most people were involved in agriculture or small-scale manufacturing known as the domestic system. In this system, women had prominent roles in both farming and domestic production. They would also be expected to juggle this with raising children.
However, this all took place before Victorian times, where the concept of "separate spheres" (that is that the man went to work and the woman stayed at home) was largely unknown. Therefore, there is a lot of evidence to suggest that children were brought up by both parents.
After 1800, factories and the process of industrialization happened in Britain and parts of France and Germany; however, this did not give people more free time. The factory system took unskilled laborers from rural areas and put them to work for as long as sixteen hours per day. Lower-income women did not have access to the goods that were being produced. Low-income children were put to work in the factories too. It was not until the Factory Act of 1833 that working hours were actually addressed. Factories were dangerous places for all workers including women and children.
By the late 19th Century, industrialization had gained momentum in Europe and much of the United States. However, the Women's Movement was still in its infancy despite the work of those who attended the Seneca Falls Convention in 1848. Middle-class women would meet regularly and discuss the problems they faced, but they were not taken very seriously. In Britain, the Suffragist Movement was largely ignored by most politicians.
At this point, it would be safe to say that industrialization had very little to do with the Women's Movement. The only relationship that can be clearly seen was the Temperance Movement, which was organized by middle-class, Christian women. This is linked to the process of industrialization because it created urbanization: more people moved to large towns and cities. Inevitably, this saw a rise in alcoholism among men as they spent much of their time and money drinking. Women were central in challenging this culture.
How does industrialization link to the Women's Movement? This can be answered more clearly if we consider the effects of industrial labor during World War I. Before the outbreak of war in 1914, women's movements were not very successful. Only in New Zealand had women gained the right to vote, in 1893. Once war started, the Suffragists agreed with politicians that they would suspend their activities and support the war effort. This involved putting lots of women to work in factories to replace the men fighting the war. Over time, the image of the weak, frail Victorian woman was replaced. More men, including politicians, recognized the capabilities of women. It is no coincidence, therefore, that shortly after the end of World War I, countries such as Britain and the United States passed laws to allow women to vote.
Therefore, the only clear link between industrialization and the Women's Movement can be seen during and after World War I, but to suggest that industrialization led to women gaining the vote is not a line of argument that historians generally take.