The phrases "the Matter of Britain," "the Matter of France," and the "Matter of Rome" seem to have been taken directly into English from Jean Bodel's verse some time in the late 19th century. Before then, "the Matter of Britian" seems to have been called by other names, such as "the Arthurian Cycle," in English.
The first time I can find "Matter of Britain" used in Google books is in The Flourishing of Romance and the Rise of Allegory by George Saintbury, which was first published in 1897.
More evidence that these phrases were taken directly from Jean Bodel's verse into late 19th century English, and were not timeworn English phrases, comes from the 1920 book A History of English Literature, by William Allan Neilson, in which we find
The great majority of the romances fall into groups or cycles concerned with what an old French poet called the “matter of Britain,” the “matter of France,” and the "matter of Rome."
Since the word comes to us from Old French, and seems to have been translated into English using the cognate word matter (whether or not this was the best English word), what we really need is a good Old French dictionary.
The Anglo-Norman Dictionary has for one of the definitions of matire:
subject matter, topic (of a literary work),
which is exactly the sense that Jean Bodel used it in. (And matire is the same word as Jean Bodel's matere – Old French spelling was as variable as Middle English spelling.)
The OED has the very similar definition (which is not surprising, since a large amount of Old French vocabulary was incorporated into Middle English after the Norman Conquest):
"The subject of a book, speech, etc.; a theme, a topic, a subject of exposition. In early use also: a narrative, an account, a tale. Obsolete."
One of their citations for this meaning is (ca. 1340)
The matere of this boke is crist & his spouse.