How did Eleanor Roosevelt change the role of the First Lady during Franklin D. Roosevelt's time in office?
She was a feminist.
Eleanor took more of an open role in the first lady. Most first ladies did much of nothing for most of American history. Eleanore Roosevelt was a great influence on many other first ladies that took more of an open role, such as Nancy Reagan, Hillary Clinton, and Michelle Obama. She influenced a lot of social programs and legislation. Essentially she made the point that first ladies don't always just have to be quiet. They can change things too.
She used the prestige of the First Lady position to advance social causes.
Few prior First Ladies had gotten involved with politics at all--Edith Wilson spoke on her husband's behalf when Woodrow Wilson was recovering from a stroke, but she was a bit of an outlier--but Eleanor Roosevelt lent the prominence of her station for social causes, such as racial equality and Universal Human Rights.
Instead of being primarily a hostess for state dinners, she used her media savvy as a columnist, public speaker and radio personality to promote both her husband's and her agendas for social progress. After FDR died, she spent seven years as an American delegate to the UN General Assembly.
Her two immediate successors, Bess Truman and Mamie Eisenhower, declined to use their position for activist purposes. Other First Ladies (notably Nancy Reagan, Hillary Clinton and Michelle Obama, and to a lesser degree Jackie Kennedy and Betty Ford) pursued the First Lady activist role with great enthusiasm.