I am not a Butt descendent so my research of the family has been focused on the connections with the Olive family of Columbia County, GA. It is my understanding that Moses, Jeremiah, John and Joshua Butt were sons of Josiah Butt of Halifax County, NC. Moses came to Georgia first, to claim his father's bounty land in Warren County. The slaves he brought with him mutineed and killed Moses. Jeremiah and John then came to Georgia to claim their father's land. As you stated, Jeremiah married Peace Willingham in Columbia County and settled in Warrenton, GA. They are buried in their family plot in the old Church/City cemetery. John married Frances Olive in Columbia County on June 20, 1810. In 1814 Frances' father, James Olive, deeded 3 slaves to Frances and John Butt and in 1820 John Butt bought 375 acres on Kiokee Creek in Columbia County, near Frances' brothers. We don't know what came of Frances after the 1820 land purchase, but nobody from the Butt family was at her father's estate sale in December of 1820.
I have not found when John married his second wife, Mary Oliver. I did jot down notes on Oliver's in Warren County since Olive has been mistaken for Oliver at times. I found no marriage record in Warren County for John and Mary, but I did make a note of Mary's fathers will. In it Benjamin Oliver refers to her as Mary Oliver, but in the settlement of his estate John Butt is listed as a legatee. I have to assume they married sometime between those 2 events, but I did not make note of those dates. (As I said, I was looking for Olive history and those two items did not seem significant to my research at the time.)
It is safe to assume that John's 2 oldest daughter, Mary Elizabeth Butt, born 1811 (wife of her cousin William Moore Butt) and Sarah A. Butt, born 1812 (wife of Montclaiborne Andrews) were Frances' children. We know that Joshua Willingham Butt was Mary's son. Without knowing when Frances left the picture and Mary entered, we can't be sure of John's other children (Oliver Perry, Adaline Portlock and John Decatur) although they were probably Mary's children. We know Montclaibourne Andrews named John Decatur Butt as his Executor and John Decatur Butt administered his sister Sarah Butt Andrews estate (as well as Sarah and Montclaibourne's daughter Frances Olive Andrews estate)and was guardian over Sarah's 3 surviving children: Warren, Mary and Laura.
As to why John named his youngest Joshua Willingham, we have to assume it was due to close family connections but no known direct lineage.