Eliza Lucas Pinckney was one of the most accomplished women in colonial history. It was due to her efforts that the colonial economy of South Carolina came to be based on the cultivation of indigo. Her experiments with growing indigo and producing dye showed people how to do those things in South Carolina's climate and soil conditions.
Pinckney's connection to republican motherhood comes from the fact that she was, herself, so accomplished. She managed plantations both before her marriage and after being widowed. She came to be very well educated, reading many of the classics from her father's library. This was in line with ideas of republican motherhood, which held that more educated women would make better mothers.
You can argue that her children proved that she was a good mother. For example, her oldest son was one of the signers of the Constitution and was twice a candidate for president. Her other surviving son was ambassador to Spain. Since their father died when they were 12 and 8 respectively, it is clear that their mother had much to do with educating them, just as ideas of republican motherhood said she should.
Eliza Pinckney was a very accomplished and educated woman whose sons became very prominent themselves. This is taken as an example of republican motherhood because it proves the idea that more educated women could do a better job of raising "republican" children who were fit to participate in a republican form of government.