“An uncontrolled time proceeding in the backward-forward direction is subject, as is known, to fluctuations during which small extensions of it (say .05 of a second) will make a measurable angle with the backward-forward direction. If, now, we suppose this increased to a right angle, this time will proceed from eckward to andward” — so the words appeared to Scudamour’s memory when he told us the story — “and will cut an ideally normal time at right angles.” 
So far as I can tell, the words “eckward(s)” and “andward(s)” are Lewis’s own inventions. I have been unable to find them in anything predating The Dark Tower (except as completely unrelated proper names). For the most part, the time-travel literature on which Lewis drew for inspiration (e.g., H.G. Wells, J.W. Dunne, Murray Leinster, et al.) either coin nothing new, making do with existing words; or else borrow terminology from the world of spatial dimensions — e.g., Leinster’s “sidewise in time”. H. Beam Piper, whose works were probably too late to have influenced Lewis, adopts “paratime”, which serves well enough — but there is something remarkable to me about Lewis’s invented adverbs, “eckward” and “andward”. They feel just right; they appear perfectly appropriate and look like genuine English words. Naturally, I was curious about their possible etymologies. Now, I have not read all of the available scholarship on The Dark Tower — far from it! — but from a quick search, it does not appear to me that anybody has written much of anything on this subject. If anyone out there knows otherwise, please pass the details along! In the meantime, I’ve had some thoughts.
Two possibilities to explain “eckward” and “andward” come readily to my mind: Greek and Old English. To tip my hand, I think it’s probably the latter, but I’ll address them both, and I welcome your input.
To make up Lewis’s words, the Greek prefixes in question would most likely come from the prepositions ἐκ, ἐξ “out of, (away) from” and ἀνά “(up)on, along, up through, thereon”, respectively. The resulting meanings would therefore be something like eckward = “in a direction away from or out of (normal time)” and andward = “in a direction along or up through (normal time)”. The first would serve us well, but the second would not; such a meaning seems opposite to what Lewis meant. The only scholar whom I have found to venture any opinion on the matter is Sanford Schwartz. He (only tangentially) advances the Greek theory in his recent book — for “eckward”, at least; I haven’t seen an opinion on “andward” . But while Greek would ordinarily be a natural assumption for Lewis (after all, the Four Loves are all given their Greek names), there are some problems here.
For one, as I’ve hinted already, the explanation for “andward” isn’t nearly as good as the one for “eckward”. And where would the d come from, if the prefix is the Greek ἀνά? For another, Latinizing ἐκ, ἐξ to “eck–” would not follow the established pattern for the use of that prefix in English (e.g., it’s eclipse, not *ecklipse or *eklipse). For yet another, the suffix, –ward, is most definitely Germanic. If the prefix were Greek, we’d have rather uneasy bedfellows from competing word-stocks. Such admixtures are not unprecedented in English, but they are not as common as unalloyed compounds. And especially in the case of a neologism by someone with Lewis’s knowledge of language (rivaled among the Inklings only by Tolkien’s or Barfield’s), one would have to expect more consistency. So what is the other alternative?
As I said, –ward is Germanic, going back to Old English –weard, ultimately from a Germanic root meaning “to turn”. Cognates to the Old English include Old Saxon –ward, Old High German –wart, Gothic –wairþs, and Old Norse –verðr. Might the prefixes be Old English along with the suffix? Certainly! If they are, the most likely sources are OE and “without, against” and éce “eternal, perpetual, everlasting” . The latter already carries a temporary meaning, so we have reason to feel optimistic! Moreover, there actually is an attested OE word andweard, which, had it survived, would have become a Modern English word of precisely the form in question, “andward”. It means “present” in the spatial sense (i.e., “without any specific direction”), but this would obviously tempt a knowledgeable word-maker to apply it to the temporal dimension as well. If this hypothesis is the correct one, then the two words would mean something like andward = “all time in the present” and eckward = “all time perpetually conceived”, both contrasted with the mundane temporal dimension we all know from experience.
This etymology fits the story very well, where the fictive time-scientists of Lewis’s Othertime explain that “[a]t the [...] moment of intersection [of two perpendicular timelines] the whole series of events in each of these times will then be contemporary to those living in the other”, and “a consciousness which succeeded in passing [from one timeline to another] would attain to endless time, and the Time Square [i.e., a two-dimensional temporal plane], though finite, would be endless or perpetual” .
I feel satisfied, therefore, that the Old English etymology is most likely the one Lewis had in mind, in spite of his more typical attraction to the Greek or Latin. What do my readers think of this hypothesis? And why haven’t other writers, of both fictional time-travel stories and nonfiction books on time, picked up these very useful words? As someone who used to speculate on temporal dimensionality quite a lot (once upon a time :), I would certainly have found them useful!
 Lewis, C.S. The Dark Tower and Other Stories. Ed. Walter Hooper. New York: Harcourt Brace Jovanovich, 1977, p. 84.
 Schwartz, Sanford. C.S. Lewis on the Final Frontier: Science and the Supernatural in the Space Trilogy. Oxford: Oxford University Press, 2009.
 An alternate possibility is OE éc “also”, with a meaning of “also-time”, but this strikes me as a bit less plausible.
 Lewis, pp. 84, 87.