Partly an offshoot of Massachusetts, New Hampshire was given a separate status in 1679.
The first JONES found in the records of Durham, N.H. is a Stephen Jones. He is listed as a "Culler of Pipestaves". Listed as an inhabitant of Oyster River, he took the freeman's oath 15 May 1672. He received land at Johnson's Creek in 1672 and was living in 1680. It appears he died before 1682 for his widow was taxed in 1682.
"Live Free or Die" it states on the N.H. quarter per above. Stephen Jones certainly would have been one of those "Old Man of The Mountain".
References: History of the Town of Durham, N.H. (Oyster River Plantation), by Everett S. Stackpole & Lucien Thompson, (c. 1913/14)
Jones Journeys, Vol. 12, 1984 - 1985, p. 875.