There seems to be no firm evidence base for the origins of the name JAUNCEY, and it is a murky area due to the phonetic similarity with O’Shaughnessy, Johns, Johnson and, for that matter, Jones. The best guesses suggest it is a French locative name, but due to the spelling similarity between JOHNSEY and John, there are speculative suggestions of it being patronymic in origin (see below); the -“y” suffix being interpreted as a possible diminutive.
The Internet Surname Database suggests Jauncey is a locative name with Medieval French origins, a corruption of CHAUNCEY: see https://www.surnamedb.com/Surname/Jauncey © Copyright: Name Origin Research 1980 – 2017
The latest edition of The Oxford Dictionary of Family Names in Britain and Ireland (2016), however, suggests a different French origin:
“From Gency in Cergy Val d’Oise or Janzé Ille-et-Vilaine France.”
Ref: Hanks, P., Coates, R., McClure, P., Oxford Dictionary of Family Names in Britain and Ireland (Oxford University Press, Oxford, 2016), available as an eBook here: The Oxford Dictionary of Family Names in Britain and Ireland
Frustratingly, this authoritative resource gives no source or evidence for either suggested origin. Gency (Normandy) is a tiny place. My research so far has not traced any significant migration from Janzé (Brittany) to England. It is not in the lists of Huguenot names, and JAUNCEY occurs in Herefordshire before the arrival of Huguenots in England.
Origins of the placename Janzé are similarly uncertain:
“Beginning with the Roman occupation of Gaul up to the 10th century, Janzé was called Janziacum. The name is thought to come perhaps from the gallo language word “jan” (a type of shrub) or from the name of the Roman god Janus or from Gennitius (a Gallo-Roman name) plus the suffix -acum.[3] The name changed to Janzéium, Janzay or Janzey and then finally to Janzé from 1216 on.[4]: 27 ” (Wikipedia, Janzé, Ille-et-Vilaine Janzé – Wikipedia)
There is no separate entry for JOHNCEY in Hanks et al (2016), but JOHNSEY is listed as a variant of JAUNCEY.
Conversely, the name JOHNCEY (with JOHNSEY as a variant) has an entry in the International Surname Database, but its claims are largely speculative and unsourced:
“Recorded in a number of spellings including Hansie, Jancey, Jansey, Jancy, Jansie, Jencey, Johncey, Johnsey, Johnsee, Joinsey, Joncey, Jonsey, and even Ginsie, Gaunsey, and Yancey, this is probably an English surname, although it could be Dutch or Flemish. Its origins are obscure and unproven, although it has been suggested that it may be the forerunner of the American descriptive word Yankee. However spelt it is almost certainly one of the many patronymics which developed from the ancient Christian name John or Johann… “ https://surnamedb.com/surname/johncey
The British Surnames website, equally unsourced, also suggests both patronymic and French origins (the key phrase here is “is believed to come from…”)
“Jauncey is a surname of British origin that has roots in medieval England. The name is believed to come from the Old French name “Jonice” or “Jonissi,” which originally meant “little John” or “John’s son.” Over time, the name evolved into Jauncey and became a distinct surname.” https://britishsurnames.uk/surname/jauncey