Abigail Adams is seen as an example of the ideals of Republican Motherhood because she tried to educate her children both morally and in terms of political and societal concerns. She also advocated for the idea that women should be allowed to get more education. She argued that this was important because women were going to become (as the ideas of Republican Motherhood said they would) the teachers of the next generation of youth. She believed that women needed to be educated in order to properly educate their offspring.
An example of Adams advocating for women's education can be seen in the following quote from the historytools.org link. Here, she is saying that educated women are the ones best placed to properly educate the young:
If much depends as is allowed upon the early
education of youth and the first principals which are instilled take the deepest root, great benefit
must arise from literary accomplishments in women.
In addition to advocating for women's education, Adams educated her own children to be good people and good citizens, just as the ideals of Republican Motherhood said she should.