The earliest known Avent emigrant to America was a Thomas Avent (you can see a lot more info on him and his life under the heading "Col. Thomas Avent" on this website). There is no known evidence of his birthplace other than the fact that we know he emigrated to Virginia from Great Britain around 1701. From the "Brunswick Co., VA, Court Order Book 1", pg. 241:

"3-May-1739 - Thomas Avent makes oath that it is now 38 years since his importation from Great Britain..." 

It's highly likely that he was from Devonshire, on the southwest coast of England. Avents have been found there and in neighboring Cornwall at least since the 1500's, and the name is not uncommon there today. Based on the webmaster's research, most Avents in that area have been found in the area bounded by Dartmoor on the north, the coast to the south, Plymouth to the west, and Brixham to the east.

*** August, 2021 update ****
The webmaster was recently matched up with a DNA relative on Livingdna (a UK based company specializing in British DNA). This connection was born in Devon from a Devon-based family and according to her records she is a 10X great granddaughter of John Avent and Suzan Luckrofte. Other information (which you can see on Thomas Avent's page in the "Avent Database" section of this website) indicates strongly that they were the great-grandparents of Col. Thomas Avent, so it is now pretty definite that Col. Thomas Avent did in fact emigrate to Virginia from Devon, England.

The following is an excerpt from the paper 'The Avant-Avent Family of England" by well-known British genealogist Peter Wilson Coldham, taken from David Avant, Jr.'s "Florida Pioneers and their AL, GA, Carolina, MD & VA Ancestors"

"It has been established through a detailed study of the sources that the Avants/Avents were mainly established from at least 1541 in Plympton, St. Mary, Devon, on which date John Avent was a witness at the Inquisition Post Mortem of Robert Brett. At later dates the Avents are found in Wembury, Brixton, Modbury, and other Devon parishes and in Cornwall. 

The first mention of an Avent in the London area occurs in the Westminster Denization Rolls of 36 Henry VIII where a Raynt Avent, born in Normandy, is listed as living in Westminster with a wife and children in 1544. The Avent name occurs spasmodically in a number of London parish registers after that date but, for the most part, it seems likely that these were emigrants from Devon and not London born. The first such after Raynt Avent (of whose family no further information has been forthcoming) is George Avent, a waterman, against whom a suit was brought by the citizens of Gravesend, Kent (will, p. 51). There is no evidence of any Avent migration from London to the West Country and such a move would seem improbable. It seems more likely that the Devon and Cornwall family took its name from a local village, possibly Advent in Cornwall. There is also no evidence to suggest that any of the Avents settled in England were of Huguenot extraction. That the separate families of Devon and Cornwall were closely related can be demonstrated from the fragmentary remains of an early legal document."

Peter Coldham's suggestion that the name derives from a local village called 'Advent', in Cornwall, is the most likely explanation for the origin of the name, but not the only one. The Dictionary of American Family Names, Oxford University Press has the following entry for 'Avent':

"English (of Norman origin): probably from a reduced form of the Anglo-Norman French personal name or nickname Avenant 'suitable' or 'handsome'.

Mr. Coldham's comment that there is no evidence that the English Avents were of Huguenot descent is interesting also. The Huguenots were French Protestants who fled France due to religious persecution, and many of them settled in England along the southern coast, which is the area in which most English Avents lived. Most Huguenots were considered to be upper-class, if not aristocratic, and there seems to be a desire among many these days to claim Huguenot descent. Many Avent genealogists have fallen victim to this desire, but there does not appear to be any evidence to support the claim, and, as Coldham points out, existing evidence seems to contradict the idea. Most Huguenot migration to England occurred after the St. Bartholomew's Day Massacre took place in France in 1572, and Avents have been documented in England well before this date. 

The webmaster has seen references to primary sources which purport to list Thomas Avent as a Huguenot, but he has never been able to locate any of these sources, and contacts at the Huguenot Society of America state that they have no record of any Huguenots named Avent. The Huguenot records that the webmaster has seen, including records of the Huguenot settlement Manakin Town in Virginia, and various passenger lists from ships transporting Huguenots to America do not show any Avents.

In a letter dated 30 September 1969, Herbert A. Elliot, Registrar of "The Huguenot Society of the Founders of Manakin in the Colony of Virginia" wrote the following to a Mrs. Travis H. Clark "I know of no proof that Colonel Thomas Avent was of Huguenot descent, in fact, all available evidence indicates that he was not."

As Coldham points out, the earliest records of Avents in Devon place them at a village called 'Plympton, St. Mary'. Plympton, St. Mary still exists in Devon, and an excellent website for the parish church there can be found here. Avents have been linked to this church for hundreds of years, and there are many Avent burials there. Another website (now gone) stated that one of the church bells (dating from 1737) was donated by 'Robert Avent - Church Warden'.

According to this website there was until recently a manor house there that had been in the possession of the Avent family for five generations. 

'The manor of Hemerdon gave name to an ancient family, who possessed it till the year 1296. In 1396, John Crocker, Esq., of Lyneham, became possessed of it by marriage with the heiress of Corim. (In the year) 1632, ...John Crocker, Esq., sold (the property) to Peter Ryder, and Tristram Avent, Esq. A portion of Hemerdon was purchased, a few years ago by George Woollcombe, Esq., who inherits from the Avents (the manors of) Hemerdon and Bickford-town. The manor-house at Bickford, which was, for five generations, the seat of the Avents, has been taken down....'

At this website for 'Devon Surnames' you will find the following quote: 

'South Devon names. Avant is Newton Abbott & Avent is Plymouth in 1881. In 1851 Avant is more Teignbridge & Avent is in the South Hams. Avent in South Hams in 1544 but Avan, Aven & Avenant all recorded then in Bishopsteignton.' 

The 'South Hams' is a region in southern Devon, along the coast, and has been described:

'The South Hams of South Devon is an area of exceptional beauty and contrast with river estuaries, rolling hills and thatched cottages surrounding the lovely old towns of Dartmouth, Totnes, Kingsbridge, Salcombe, Modbury and Ivybridge and pretty villages that abound in the South Hams district of South Devon.'

Not all Avents in America are necessarily descendants of Thomas Avent, though. The webmaster was recently perusing some Revolutionary War records and ran across several Avents living in CT and NY, who were almost certainly not descendants of Col. Thomas Avent of VA. The 1800 census shows a Josiah Avent, of Winchester, Litchfield CT, who also was almost certainly not a descendant of Thomas Avent. There was a George Avent who emigrated from England in 1861, settled in New York City and fought in the Union Army during the War Between the States. Each of these Avents likely has descendants living today, none of whom are descended from Col. Thomas Avent of VA.

Also, a common variant of the Avent name is 'Avant'. The webmaster has looked into a couple of Avant lines, and they could be traced back to Thomas Avent, but there is a large Avant family in SC, and many Avant genealogists believe that they descend from a Huguenot named Caleb Avant, who settled in Prince George Winyah Parish, near Georgetown SC, in the early 1700's. There are also claims that a Francis Avant recorded a land grant in SC in 1709. Neither Caleb nor Francis were descendants of Thomas Avent of VA. Nevertheless, the webmaster has researched numerous Avent families and, with few exceptions, most were provably (or very likely) descended from Thomas Avent of Sussex Co., VA.