"Tudor" suggests, in Lewis's history of England as shown in this novel, a representative of Logres as opposed to Britain.
The fantastical theory of English history which is described in That Hideous Strength, and which perhaps in some part reflects Lewis's own views on the history and culture of this country, is described most clearly in Chapter 17, part IV:
"It all began," he said, "when we discovered that the Arthurian story is mostly true history. There was a moment in the Sixth Century when something that is always trying to break through into this country nearly succeeded. Logres was our name for it--it will do as well as another. And then . . . gradually we began to see all English history in a new way. We discovered the haunting."
"What haunting?" asked Camilla.
"How something we may call Britain is always haunted by something we may call Logres. Haven't you noticed that we are two countries? After every Arthur, a Mordred; behind every Milton, a Cromwell: a nation of poets, a nation of shopkeepers; the home of Sidney--and of Cecil Rhodes. Is it any wonder they call us hypocrites? But what they mistake for hypocrisy is really the struggle between Logres and Britain."
He paused and took a sip of wine before proceeding.
"It was long afterwards," he said, "after the Director had returned from the Third Heaven, that we were told a little more. This haunting turned out to be not only from the other side of the invisible wall. Ransom was summoned to the bedside of an old man then dying in Cumberland. His name would mean nothing to you if I told it. That man was the Pendragon, the successor of Arthur and Uther and Cassibelaun. Then we learned the truth. There has been a secret Logres in the very heart of Britain all these years; an unbroken succession of Pendragons. That old man was the seventy-eighth from Arthur: our Director received from him the office and the blessing; to-morrow we shall know, or to-night, who is to be the eightieth. Some of the Pendragons are well known to history, though not under that name. Others you have never heard of. But in every age they and the little Logres which gathered round them have been the fingers which gave the tiny shove or the almost imperceptible pull, to prod England out of the drunken sleep or to draw her back from the final outrage into which Britain tempted her."
In this story, Logres is the ancient nation of King Arthur, of poets and romance, represented in the modern day by Ransom's little community at St. Anne's, while Britain is the modern nation of wars and empires, represented in the modern day by the N.I.C.E. at Belbury. England is described as "swaying to and fro between Logres and Britain", and all the events of this story, the rise and fall of the N.I.C.E., as a time when "Britain rebelled most dangerously against Logres and was defeated only just in time".
To see how this ties into Jane's surname, it is necessary to venture outside of the text of the story and look at Lewis's mention of the Tudors in another context, his letter to a fan dated 26 January 1954:
We wouldn't call Alfred and Egbert and all those the "British" line. They are the "English" line, the Angles, who come from Angel in South Denmark. By the British line we'd mean the Celtic line that goes back through the Tudors to Cadwallader and thence to Arthur, Uther, Cassibelan, Lear, Lud, Brut, Aeneas, Jupiter. The present royal family can claim descent from both the British and the English lines. So, I suppose, can most of us:
In this paragraph, although his usage of the word "British" here has different connotations from its usage in That Hideous Strength, Lewis is clearly describing the Tudors as part of the line of King Arthur. In the mythos of That Hideous Strength, this would put the Tudors firmly into the Logres camp, which fits thematically with Jane's maiden name.
It may even be the case that Jane is literally descended, somehow, from the Tudor monarchs. Merlin said explicitly that her (and Mark's) ancestry was significant and their union had been planned for centuries:
"For a hundred generations in two lines the begetting of this child was prepared; and unless God should rip up the work of time, such seed, and such an hour, in such a land, shall never be again."
And when Jane's maiden name is mentioned for the only time in the story:
"What was your maiden name?" asked Miss Ironwood.
"Tudor," said Jane. At any other moment she would have said it rather self-consciously, for she was very anxious not to be supposed vain of her ancient ancestry.
This seems to indicate that Jane really does have "ancient ancestry" as the name suggests.