I believe your question tackles two issues, that of the place name and that of the national identity of the British people. I will address former portion of that question first as I find it to be a somewhat easier question to answer.
When did the term Britain come into use?
The Greeks: The earliest existing records of the word are quotations of the Pytheas Periplus [a 4th century source by Pytheas of Massalia of which no written copies survive] by later authors, such as those within Diodorus' Bibliotheca historica (c. 60 BC to 30 BC), Strabo's Geographica (c. 7 BC to AD 19) and Pliny's Natural History (AD 77).According to Strabo, Pytheas referred to Britain as Bretannikē, which is treated a feminine noun. Although technically an adjective (the Britannic or British) it may have been a case of noun ellipsis, a common mechanism in ancient Greek. Other descriptions of Britain appear in a myriad of other sources from Aristotle {source here} who used nēsoi Brettanikai, Albion and Ierne to refer to the island group, Great Britain, and Ireland, respectively, to Marcian of Heraclea in his Periplus maris exteri, described the island group as αί Πρεττανικαί νήσοι (the Prettanic Isles where you can already begin to see the connection to the modern place name and pronunciation).
Early Medieval Britain and the Anglo Saxon World: As you well know, the collapse of Rome led to the extreme contraction of its territories as well as its sphere of knowledge. The ancient descriptions by the Greeks and Rome's own word for the Isle ( Brit(t)an(n)ia )) fell out of use. The languages that developed in Sub-Roman Britain did not have a word for Britain like their predecessors, and the practiced lapsed (at least in written form) until Alfred the Great, where the term reemerged in Old English. The Graeco-Latin term referring to Britain entered in the form of Bryttania, as attested by Alfred the Great's translation of Orosius' Seven Books of History Against the Pagans (Read more about it here ). While the Latin name Britannia re-entered the language through the Old French Bretaigne. The use of Britons for the inhabitants of Great Britain is derived from the Old French bretun, the term for the people and language of Brittany, itself derived from Latin and Greek, e.g. the Βρίττωνες of Procopius. It was introduced into Middle English as brutons in the late 13th century. These gradual introductions and removals of the word from our sources reflect some of the clashes within both the continent and Britain at the time. Movement of Germanic peoples, Scandinavia and later the Normans meant that language was constantly in flux.
British as a point of self reference?
Here we come into the more prickly portion of your question which has to do with self identification among the common peoples. The term "Britain//British" itself actually refers to people identifying as both English, Welsh and Scottish (all of the nations which exist in the modern isle of Britain). When did the idea of a supranational identity appear rather than simply a national identity? There is significant debate when this occurred but it likely did not occur until the 18th century following the union of the English and the Scots in 1706/1707 when the Kingdom of Great Britain was officially established. These nationalistic ideas were only furthered in the following century and a half, with wars in America as well as with the French (think Napoleon) solidifying a national identity.
By 1066 (the date your question references) the idea of England and a common English history was just beginning to develop, the kingdoms of Mercia, Wessex and Northumbria having been united less than 2 centuries earlier under King Aethelstan. The idea of an united English people had only been conceived two generations under his grandfather Alfred the Great.
TL;DR: The term Britain has been used for millennia to refer to the island of Britain, but the idea of a united British people did not truly appear until the unification of the Scots and the English in 1707.
Additional Sources and References:
Colley, Linda (1992), Britons: Forging the Nation, 1701–1837, Yale University Press,
Athelstan (c.895 – 939), BBC,
A History of Britain TV Series by Simon Schama: here
Oman, Charles (1910), "England Before the Norman Conquest", in Oman, Charles; Chadwick, William (eds.), A History of England, vol. I, New York; London: GP Putnam's Sons; Methuen & Co, pp. 15–16,
Strabo's Geography Book IV. Chapter IV. Section 1 Greek text and English translation at the Perseus Project.