In investigating this subject, I first took a shot at getting a home run by looking at "The Straight Dope". This is regarded by many as the premier "notes and queries"-type column. Visiting its archives page and searching for "alphabet" doesn't reveal that the omniscient Cecil Adams has tackled this one - or, at least, if he has, it isn't available in his archives. The closest we get is the second half of this, about the differences between I and J and between U and V. No good.
So to more conventional Googling. The first search to bear fruit was for "alphabetical order" history origin letters, leading to a three-page article on The Earliest Evidence of an Alphabetical Order. That's the what, but not the why. Already you can see there is a reasonable degree of similarity between the alphabetical order suggested there and that of the Greek alphabet. A key resource listed in the bibliography of that resource was "The Alphabet Labyrinth", by J. Drucker in 1997 - but more of that later.
Another interesting result from that Google was found on the instructions for Mirror History Dictionary, a project which attempts to categorise words from many languages. The author of that site cannot offer a definite conclusion, but gives the conclusion that (s)he has spent time thinking about it. The best (s)he offers is "Is it possible that the Greek and Hebrew alphabets originally told a story? This becomes a possibility when you consider that each letter originally had a name attached to it. In essence, a word! In that case the arrangement of letters might illustrate nothing more than the order of words in a sentence, or a story." Not a good sign that progress is probable.
So we return to looking for "The Alphabet Labyrinth". A search for this reveals a very disappointing three hits and no record on Amazon. However, one of them is this page on alphabet origins (beware: on FortuneCity and so pops up ads, Gator launchers and the like at a disgusting rate on Windows MSIE). About a fifth of the way down the page is a section on Ugaritic (the language of the Syrian coastal city of Ugarit) and Letter Order. I can do no better than quote the best answer I have been able to find to the original question.
"There is a remarkably difference between the South Arabian tradition and the West Semitic: the letter ordering. The South Arabian alphabet has the order h, l, h, m, etc..., while West Semitic has the order ', b, g, d, etc.
No one is sure why those particular sequences of letters. Maybe it is some mnemonic device that we no longer understand."
Author Steve Bett goes on to speculate that "I believe that the early alphabets and proto-alphabets had a reference in addition to a name, a sound, and a shape. For some reason, no one has bothered to catalog the references or possible references of the letter shapes. If we are dealing with acrophonic phoneticized pictograms, these will be language specific. You can't move a picture name across a linguistic boundary any better than you can a number name. If we can identify what the phonogram shape is a picture of, then we might be able to make some sense out of the sound assignments in Linear B, the Byblos Syllabary, hieroglyphics, and the Semitic letter shapes that don't seem to have a reference."
This is the best answer I have been able to find to the question, but not wholly satisfactory.
A search for "The Alphabet Labyrinth" on abebooks.com, the premier second-hand books reference, suggests that part of the problem may be that we have been looking for the wrong title. The book referred to above is actually called The Alphabetic Labyrinth: The Alphabet in History, Mysticism and Imagination and remains in print in paperback, ISBN: 0500280681. £16.95 from its British publishers, Thames and Hudson. (A more expensive alternative, possibly considered inferior, is The Alphabet: A Key to the History of Mankind. ISBN: 8121507480.)
Amazon will let you look at some of the pages from "The Alphabetic Labyrinth" - looking at the contents page does not make it particularly clear to what extent the book covers the invention of alphabetical order and the index makes it no clearer. On the whole, I get the impression that Professor Johanna Drucker approaches the issue from the visual art side of things, rather than the ordering side of things. Certainly this appears to be the case based on the syllabus of the course she taught in Spring.
Of course, you could always send her e-mail and ask...
Well, this took rather more like two hours than one and didn't come up with a particularly satisfactory answer, but it was fun. If I were to spend another hour or two on the topic then I would concentrate on the changes in alphabetical order from alphabet to alphabet. However, the links above should give you more information on the latter half of the subject.
It also strikes me that I probably could have got about as good an answer by sticking a little payment up for grabs on Google Answers and waiting. This would have cost me US$2.50. Evidently I value my time at somewhat less than US$2.50/hour. :-)
In general: yesterday was a hard day, but it did pick up towards the end. Overnight, I had a very 1995 dream: Frank Bruno won the World Heavyweight Boxing Championship and I was playing even-then-a-classic C64 game Wizball. (Shades of worrals's dreams, indeed.) Today has been rather easier, largely having been distracted by helping Dad with some manual work - monkeying about on the roof of the garden shed.
A little puzzle news: Wei-Hwa Huang only finished twentieth in the World Puzzle Championships, but was apparently very ill at the start. It happens to us all, but the illness that knocked Wei-Hwa down to 20th must have been far worse than that which knocked me up to 72nd. (I am terrible at coping with even a slight cold. Ask WPC 2001 roommate ericklendl.) Oh, and PQRST 3 runs from the 12th to 19th of October. The prospect is for ten interesting, difficult but solvable puzzles.