As can be seen in the adjunct Timeline, there were many "events" in French history that played some part (minor or major) in the establishment of Carolina. When Jacques Cartier first explored the St. Lawrence River (1519-1522) France immediately became a "player" on the New World stage. Thankfully, France was more interested in "fur trading" than they were in establishing a big contingent on North America.
Jean Francois de la Rocque, sieur de Roberval brought the first colonists to what is now Canada in 1542. Luckily for the English, this colony was a total disaster - it was another sixty (60) years before the French attempted colonization again.
So, why did it take so long?
Quite simply - wars. France was soon at war with the Hapsburgs (1521-1559), then the Protestant Reformation brought on the Wars of Religion (1563-1598).
The year 1598 brought with it about twenty (20) years of relative peace, and the French took advantage of this by soon sending over two more explorers with instructions to begin colonization. In 1605, Pierre du Gaust, sieur du Monts settled Port Royal. In 1608, Samuel de Champlain settled Quebec. Both of these small colonies were primarily interested in trading with the native Indians for beaver furs - the latest rage in Europe was anything with a beaver fur collar - and, Russia had just increased their fur prices.
The peace didn't last long enough. In 1618, France was caught up in the Thirty Years War - a "subset" of the Eighty Years War with the Spanish domination of The Netherlands, which had started in 1568. Both England and France had tried to stay out of this conflict, but by 1618 both were deeply involved, and neither could extricate themselves until the Spanish eventually conceded and granted The Netherlands their independence in 1648.
In parallel with the Thirty Years War, France was once again embroiled in a religious war - the Huguenot Rebellion of 1625 to 1628. Right after that war came the Franco-Spanish War of 1635 to 1659. During these wars, France did manage to send over a third group to Canada that settled Montreal in 1642.
By now, the Dutch were fairly entrenched along the eastern seabord in New York, New Jersey, Delaware, and Pennsylvania. The English were firmly established in New England and Virginia, and the Spanish simply "owned" everything from Florida to California. But, apparently the Dutch, the English, and the Spanish had no great interest in the Mississippi River basin, the Ohio River basin, nor much else.
So, in 1673, the French explored what is now known as the Louisiana Territory and soon thereafter began settlement in earnest. Eventually, the French wrested Mobile Bay and what is now Louisiana away from the Spanish, and New Orleans was founded by the French in 1718. The French had a major "foot-hold" in North America, one to rival the English along the East Coast, which had by now wrested everything away from the Dutch.
Way back in 1562, Jean Ribaut attempted the first French colony in Carolana - he called it Charlesfort. He even coined the name of the future colony - Carolana - latin for Charles, after his French king - King Charles IX. Later, when the English took over the land, King Charles I liked the name and selected it himself for the new English colony in 1629.
Ribaut's Charlesfort was a "flash in the pan" for the French. Not long after the French arrived and barely built a few structures, the Spanish sailed into Port Royal harbor and destroyed Charlesfort. Of course, this was only revenge for the Spanish since Jean Ribaut had attacked several Spanish settlements in Florida on his way to Carolana in 1562.
From 1562 to 1702 the French left Carolana alone. They were focused on their Canadian settlements and their Lousiana Territory. Carolina was no threat to them, even when the colony finally began to grow in the late 1600s.
But - war once again brought change. In 1701, The War of Spanish Succession brought France, England, Spain, and Austria into yet another major conflagration, one that spilled out of Europe into the New World. England was now at odds with both France and Spain, both with significant interests in North America. Most colonial action took place elsewhere, but there were several "events" in Carolina.
In 1702, the French decided to attack Charles Town. Somehow, the Carolina colonists found out about the pending attack before it arrived and they were ready for the French. The small fleet that sailed into Charles Town harbor was handily defeated and the French went back to New Orleans, licking their wounds.
For the duration of this war (1701-1714), the French sent "privateers" out of New Orleans to attack all English colonies along the eastern seabord, and all parts of Carolina were hit at one time or another. Very little damage came from these petty skirmishes.
During the course of their exploration and settlement of the Louisiana Territory the French managed to establish fairly good relations with the native populations they encountered. Unlike the Spanish, the French did not consider "using" the Indians to attack competing colonies - until much later - of course, they figured this tactic out by the 1750s when the French and Indian War broke out - but, this is much later than the time-frame for early Carolina.
Also unlike the Spanish, the French "nationals" had virtually no "cultural influence" on the colony of Carolina. Even most of the French sounding "place names" did not originate due to France. One good example - Beaufort - both in North Carolina and South Carolina - were NOT named for anything French - they were named for Henry Somerset, Duke of Beaufort in England.
With that said - there were French settlers in Carolina - as early as the 1680s. The French Huguenots that escaped persecution and made their way to England soon found passage to the New World, thanks to William & Mary and Queen Anne. The Huguenots settled in what is now considered to be both North Carolina and South Carolina and they became productive members of the Carolina society. Most, however, chose to become fully integrated into the "English" culture and intermarried with those from the British Isles. Within twenty (20) years of arriving in Carolina, most French Huguenots had adopted anglicized names, or variations thereof, as well as most English customs.
So, the "influence" of the French was very minimal in Carolina. Now you know why.