Early news reports about the organization
In support of the "by 1867" origin of the secret society, here are two mentions of "Kuklux Klan" from the Pulaski [Tennessee] Citizen, published in that year. First from the Pulaski [Tennessee] Citizen (March 29, 1867):
WHAT DOES IT MEAN?—The following mysterious "Take Notice" was found under our door early yesterday morning, having doubtless been slipped there the night previous. Will any one venture to tell us what it means, if it means anything at all? What is a "Kuklux Klan," and who is this "Grand Cyclops" that issues his mysterious and imperative orders? Can any one give us a little light on this subject? Here is the order:
"TAKE NOTICE.—The Kuklux Klan will assemble at their usual place of rendezvous, The Den," on Tuesday night next, exactly at the hour of midnight in costume and bearing the arms of the Klan.
By or of the Grand Cyclops.
And then from "Kuklux Klan," in the Pulaski [Tennessee] Citizen (April 5, 1867):
Another one of those mysterious communications from the Grand Cyclops of the Kuklux Klan found its way under our door yesterday morning, left there we suppose by the Grand Turk. We are warned not to make an effort to find out objects of the "mystic klan," and to allow the Grand Cyclops to issue his orders without molestation. Well, old Cyclops, just issue as many orders as you please, but if we catch your Grand Turk "cyphering" round our door late at night, we'll upset him with a "shooting-stick." Look out, old Turk, we are some Cyclops ourself on our own premises. But we give our readers the benefit of the communication:
EDITOR OF CITIZEN:—You seem to express, in your last issue, some surprise and curiosity in regard to the Kuklux Klan, whose boldness and affrontery should so startle you. That they should dare send forth their imperial edicts; or that the GRAND CYCLOPS should presume to dispatch his Grand Turk with orders to his faithful followers, or that they should dare come so near your editorial sanctum, as to leave one of their orders under your door. ...
But seek not to know the object and designs of the "Mystic Klan," or to impeach the authority of our GRAND CYCLOPS to issue his mandates, for your efforts will be fruitless. If you see proper to publish our orders and will do so, we thank you, but more of the "Kuklux Klan" you cannot know. By order of the Klan, G.S.
Subsequent articles about the Klan—often introducing further messages from the organization—appeared regularly at the Pulaski Citizen for many weeks thereafter, published on April 12, April 19 (featuring a visitation from a personage who "appeared to be about nine feet high, with a most hideous face, and wrapped in an elegant robe of black silk" and carried "a magic wand" in "gloves the color of blood"), April 26, May 3, May 17 (a brief notice complaining that the editor had not heard from or been visited by a Klan representative in two weeks), June 7 (account of a midnight procession), June 14 (first quasi-political statement from the Klan, hinging on a joke about political versus gastrointestinal irregularity), and so forth.
The June 21, 1867, installment juxtaposes a fairly innocuous note that "Alla Hassan, having been found guilty of a gross violation of the rules of the 'Klan' and the orders of the Grand Cyclops, and appearing in his august presence in a somewhat intoxicated condition, is forever expelled from the "Klan," and deprived of all its benefits and privileges," with a longer story headed "A Diabolical Lie" reporting that unnamed complainants were reporting to "the Bureau authorities" (presumably the local branch of the federal Freedmen's Bureau) on various outrages against freedmen in Giles county, including a lynching and various assaults committed by "a [white] mob" and by a "[white] party of roughs." The newspaper asserts that the complaints—variously "diabolical lies" and "atrocious lies"—were politically motivated and utterly false. So the Pulaski Citizen's editorial outlook is clearly strongly anti-Reconstruction (as was the Klan's, when it emerged as a powerful political force in the U.S. South).
The source of the name
The newspaper shows no interest in the meaning or etymology of the name kuklux klan, and very little journalistic curiosity as to the group's motives or intentions. In fact, the ornate, rather overblown letters from the Grand Cyclops quoted in the newspaper read very much like the work of a typical mid-nineteenth-century newspaper editor.
It is not impossible that the name has some dread and occult significance, but I incline to the theory that kuklux is nothing more than an alliterative nonsense word preceding the operative word klan. However, I would be remiss not to note that the Greek word Κυκλωφ (Kuklops) might be pronounced by a Southerner speaking Greek as a third, fourth, or fifth language as something similar to "Kuklox"; the only thing giving a hint of plausibility to this idea is the fact that in Pulaski, Tennessee, the personage comparable to the later Grand Dragon or Grand Wizard of the organization was initially called the Grand Cyclops, implying perhaps the organization's all-seeing eye. Wikipedia has a lengthy article about the Klan's titles and vocabulary.
Still, anyone attempting to interpret the meaning of Kuklux should bear in mind the strain of exotic Orientalism characteristic of the early Klan's early presentation of itself (black silk or crimson robes, titles such as Grand Turk, and individual names such as "Alla Hassan"), which might invite further objectively silly foreign-sounding word inventions of the Gilbert and Sullivan school.