19th CENTURY HISTORIANS
The term Hundred Years' War originated in the early 19th century.
The Hundred Years War has become the established name for the
Anglo-French conflicts that happened between 1337 and 1453. Although
the designation does not refer to an actual event—the term was first
used in France in the early 19th century — it usefully emphasizes the
insoluble nature of the hostilities.
Noted 19th century French historian Jules Michelet's Precis de l'histoire de France (1833) uses 'la guerre de cent ans', but also refers to the war as 'la guerre des Anglais'. He divides the war into two periods, 1328 to 1380 and 1380 to 1453, 1380 being the year the French crown passed from Charles V to his 12-year-old son Charles VI. Michelet's Histoire de France was translated into English in 1847 by G. H. Smith but it does use the term Hundred Years War. However, the 1882 edition (also Smith) uses a hundred years of war.
Among English historians, William Longman in The history of the life and times of Edward the Third (1869) comes close with "a war, which never entirely ceased for a hundred years." Longman also used 'the French war'. A few years later, in A Short History of the English People (first published in 1874), John Richard Green used 'The Hundred Years' War'. This appears to be the first use in English.
CHRONICLERS AND PRE-19th CENTURY HISTORIANS
The most important chronicler of the first 'half' of the war was Jean Froissart who covered the years 1326 to 1400. In his prologue, he refers to the conflict thus:
the great marvels and the good deeds of arms that happened during the great wars of France and England and the neighboring kingdoms
Often, though, Froissart just uses 'the wars'. Other chroniclers used similar phrases, but it's important to note that their chronicles cover several wars. For example Jean de Wavrin, whose work goes up to 1471, mentions 'wars in France, England and Burgundy'; Jean Le Bel whose chronicle was heavily borrowed from by Froissart, usually just mentions 'the war' or 'the war between _____ and _____'.
Written about 100 years after the end of the war during the Tudor period, the 1555 edition of Anglica Historia by Polydore Vergil, one of several historians who has been called 'The Father of English History', uses phrases such as 'the war in France', 'the war against the King of France' and 'the great war in France'. Two hundred years later, the Scottish historian David Hume's History Of England, vol. 2 (published in 1762) used similar phrases (e.g. the war with France).
The complexity of the conflicts of the latter part of the war (1400 to 1440) is well-reflected in one of the title pages of mid 1800s English translations of Enguerrand de Monstrelet's Chronicle.