I don't have anything on these particular Achesons, but I have another Acheson family who lived at one time in Crawford Co., PA, and later in Warren Co., IL, which is adjacent to Mercer, so I can't help but think they were related.
This is a rather complex family because of several first-cousin marriages. Here is some of it:
John Acheson, b. 1710 in N. Ireland; m. 1733 to Catherine Clark; immigrated about 1735 to Little Britain, Lancaster Co., PA, 1733; died about 1778. Catherine moved with some of her children to Washington Co. PA. Seven children including sons Matthew and Humphrey.
Matthew Acheson (son of John), b. 1734 in Ireland; immigratead with parents about 1735; m. 1761 to Jean Reed; moved to Mt. Pleasant, Washington Co., PA, 1777; died 1814. Six children including sons David and Robert.
Humphrey Acheson (son of John), b. 1740 in Little Britain, PA; m. 1764 to Rebecca Reed (sister of Matthew's wife Jean); moved to Mt. Pleasant, Washington Co., date unknown. Twelve children including daughter Ann. Rebecca died about 1790 and Humphrey married widow Hull. He died 1814.
David Acheson (son of Matthew), b. 1763 in Little Britain; moved with family to Mt. Pleasant, PA, in 1777; married 1777 to his cousin Ann Acheson (daughter of Humphrey); moved about 1811-12 to Muskingum Co., OH, where Ann died in 1820. Nine children including son Humphrey. David was married second to Mary Smiley. He died in Bloomfield, OH, in 1829.
Robert Acheson (son of Matthew), born 1780 in Mt. Pleasant, Washington Co., PA, died July 20, 1848, in Bloomfield, OH; married Martha Ramsey. Daughter Mary.
Humphrey Acheson (son of David), born 1793 in Washington County, PA;married 1810 to Mary Laughland (or Laughrey), who was born in County Antrim, Ireland. He was a teacher and farmer, and served in the military during the War of 1812. At some time between 1812 and 1816 they moved to Crawford County, PA, and between 1828 and 1831 they moved again to Muskingum County, OH. Mary died there in 1843. Eight children. On January 21, 1845, in Muskingum County, OH, Humphrey married his first cousin Mary Atchison. Mary was the daughter of Humphrey’s uncle Robert Atchison and Martha Ramsey. They appeared in the 1850 census of Muskingum County, then apparently that same year moved to Simmer Township, Warren County, IL, where Humphrey died in 1861. Mary then returned to Muskingum County, where she died on July 11, 1870.
I have more on some of these folks, if you think there might be a connection with your family.