This is a good question, and I had to look up the definition of the word when I first read To Kill a Mockingbird. After moving in with her brother Atticus shortly before the beginning of the Tom Robinson trial, Scout's Aunt Alexandra quickly settled in to the social life of Maycomb. In addition to her Missionary Society meetings, Alexandra became the secretary of the Maycomb Amanuensis Club. Amanuensis has many meanings (in Latin, it roughly translates to "manual labor or laborer"), but it generally refers to someone who takes dictation or shorthand. So, we can assume that the Amanuensis Club is a group of women who are (or have been) secretaries--and Alexandra is the secretary of the secretaries.
Wow. This is a very specific question. There is one reference to the amanuensis club in the whole book. It occurs in chapter thirteen where there is a description of Aunt Alexandra and her new life in Maycomb.
She was a part of the missionary society as well as the amanuensis club. The word amanuensis is an old fashioned word, which speaks of a time when people wrote down the dictations of another person. In light of this, Aunt Alexandra was a women of letters, and she worked in a society that wrote things down for others.
The wealthy may have employed an amanuensis. Also amanuenses may have helped others who could not write. This make sense as not all in Maycomb could read and write. If we recall the black community, and Jem's and Scout's experience at Calpurnia's church, there were many who could not read and write. So, there was definitely a need for an amanuensis society.
Moreover, Aunt Alexandra was very outgoing and she loved to get involved in many activities.