It appears he was named after his Uncle, his father's older brother. Many males in the family/families have middle intial J. and after seeing Andrew J. I suspect it is for Andrew Jackson and the the males may then have the Jackson name too.
He married Caroline E. "Carrie" Warner. This couple did not have any children according to the census records. Was a farmer as can be seen by census records. It appears he stayed in the household of his mother-in-law with the family per census of 1900. By 1910 he is shown head of house. So it appears that Aaron lived all of his life in New Jersey and spent that time farming as a labor for someone, or for his own.

Source Citation: Year: 1860; Census Place: Southampton, Burlington County, New Jersey; Roll: M653_684; Page: 361; Image: 365. enumerated July 13, 1860
Saml P Hames 53
Martha Hames 40
Lydia Hames 37
*Aaron Bozarth 20 NJ
Thomas King 12
Agnes Edmond 9
Achsah Cline 9
Caleb Budden 9
**************
Source Citation: Year: 1870; Census Place: Springfield, Burlington County, New Jersey; Roll: M593_857; Page: 649; Image: 147. enumerated Aug. 20, 1870
listed in household of John J Sutton and wife Hannah
Bozarth, Aaron 30 farm labor NJ
Bozarth, John 21 labor NJ
**************

These census records are meant to identify this A J Bozarth. Also to help clarify between him and Uncle.

Source Citation: Year: 1900; Census Place: Lumberton, Burlington County, New Jersey; Roll: T623 957; Page: 5A; Enumeration District: 21. enumerated June 7, 1900
Mary Warner hd f w Oct 1816 83 widow children 6/4 PA PA PA read/write own free farm
Beulah D or I Warner dau f w Feb 1840 60 NJ NJ PA
Georgiana Warner dau f w Apr 1842 58 single NJ NJ PA
Ann Emma Warner dau f w Nov 1849 55 single NJ NJ PA
Aaron Bozarth son in law m w June 1838 61 M 16y NJ NJ NJ
Carrie E Bozarth daughter f w Dec 1847 52 M 16y children 0/0 NJ NJ PA
Samuel Sharp servant 35
*************
Source Citation: Year: 1910; Census Place: Lumberton, Burlington County, New Jersey; Roll: T624_871; Page: 5B; Enumeration District: 65; Image: 762. enumerated April 19, 20, 1910
Aaron J Bozarth hd m w 70 M NJ NJ NJ farm manager, home farm owns free
Caroline E Bozarth wfe f w 62 M. NJ NJ PA
Beulah D Warner sister in law 70 single NJ NJ PA
Georgiana Warner sister in law 68 single NJ NJ PA
Emma M Warner sister in law 65 single NJ NJ PA
Margaret Booz Aunt f w 90 widow PA PA PA
**************

By one of the persons taking a photo of the monument.
Their suggestion:
-----------------
I have a good picture of this stone and his birth year was 1839. I know the dates can be hard to see. Think though, he couldn't have been on the 1860 census if he was born after it. The death year is 1914. He was 74 when he died.