I have been unable to locate any information relating directly to the origins of the name
Cunneyworth. Burke's Peerage in England did not have any record of the name in 1995.
I looked up "Cunningham" in a book of names and found that it was originally derived from "Coneyham". Coney, or cony, was an old english word for rabbit or the fur of a rabbit (from the Latin cuniculus: rabbit, burrow, underground passage or mine), and "ham", of course, referred to a hamlet. Together, it implies that Coneyhams were from a town where the rabbit business was prominent.
The 1970 Oxford Shorter English Dictionary provides related terms conyger, conynger, cunningaire and conygarth... which all mean rabbit warren.
The same dictionary defines "cunne" as the verb "To learn to know", whence: a) To prove, test, try; b) To study.
The surname.com web site provides an alternative origin for the name Cunningham as a Scottish / Irish Place / Patronymic Name: Cunningham is a polygenetic name (meaning it has more than one source). Cunningham is a Scottish place name that described the man from the location near Kilmarnock and first recorded in 1153 as Cunegan, a word with Breton origins. The spelling with -ham added has its earliest known mention in 1180. When of Irish origin, Cunningham is the Anglicized form of O' Cuinneagain , meaning 'descendant of Cuinneagain' a personal name derived as a diminutive form of Conn = leader, chief. Cuninghame, Cuningham, Cunninghame, Coningham, Conyngham are variations of the Scot version. Conaghan, Cunnigan, Cunihan, Cunnahan, Kennigan, Kinnegan, Kinaghan, Kinnighan , and Kinihan are variations of the Irish form.
The same source states that the name Conner is derived from Middle English connere, cunnere = inspector, from cunnen = to examine, from Old English cunnan = to know. It was the occupation of the man who inspected for standards, including weights and measures.
In an email posted to RootsWeb's Yorkshire genealogy mailing list, Carole Clyde notes a reference to Cuni Onlaf: (that is, King Onlaf) in "The Annals of Yorkshire", a book by John Mayall published in 1864, containing Yorkshire history and names... so the first part of the name might mean anything from rabbit skinner to royalty.
"Worth", in addition to its usual connotation of value, was also used as early as 1575 in old english as the second part of some place names to mean "an enclosed place; a homestead".
Related names Coneyworth, Conyworth, Conneyworth, Connyworth, Cuniworth, Coniworth, Coniwoorth, Cunniworth, Coneworth, Cunsworth, etc. have been found in various early records in England, including numerous variations in East Yorkshire, in Cambridge, Buckingham, Gloucester, Lincoln, Warwick and other locations. The online LDS web site lists records dating back to the 1500s for these spellings and others, including Cunnyworth, Cunneworth, Cunnesworth, Cunningworth, Cunningsworth, Cunyngwath, Cunworth, Cunnysworth, Conygworth, Coningworth, Counewurth... and even exact matches with the spelling "Cunneyworth". The vast majority of these references are located in Yorkshire. I have also located many references to the name Conworth, including a Conworth cemetery in the county of Brant, Brantford township, Ontario. (OCFA reference: BT-085)
There are towns in central England called Coningsby (WNW of Nottingham) and Conisbrough (NW of Sheffield in South Yorkshire), and in northern England called Coneysthorpe (SE of Scarborough in North Yorkshire), Coneythorpe (NNE of Leeds in North Yorkshire), Coniston (two of them: one NW of Kingston Upon Hull, another NW of Lancaster in Cumbria), Coniston Cold and Conistone (both WNW of Leeds in North Yorkshire).
I've found several documented references to deceased Cunneyworths in the US. All the living Cunneyworths I have located are in (or from) Canada, with the exception of a limited number (Cunneyworths and Coneyworths) in England. A letter (dated August 9, 2000) from Brigham Coneyworth in Yorkshire, England, however, assures me that there are many living Coneyworths I have not yet had the pleasure of identifying. He also informs me that the local pronounciation of the name "Coneyworth" is, in fact, "Cunneyworth".