From
Alpha Chi Omega's website:
Alpha Chi Omega's Founders chose "Alpha," the first letter of the Greek alphabet, because they were forming the first fraternity in the school of music. Since they thought they might also be founding the last such fraternity, "Omega" seemed appropriate. "Kai," meaning "and," was added to form the beginning and the end. "Kai" was soon changed to "Chi," a letter of the Greek alphabet.
Also, you give Phi Beta Kappa as an example, but the meaning of those letters was secret at first.
Of course, many groups have an open (public) meaning and a closed (secret) meaning for their letters.
I do know of one GLO that used to state publically what the meaning of its name/letters was -- it could be found in some of their publications and even in at least one edition of
Baird's. But for quite a long time now, that group has not published that information anywhere -- it seems to now be considered secret. So I ain't telling which group it is.