It seems like the Germans in World War I were the first to use Operation names, and the practice orginated towards the end of that war. According to one source Operation Names were both "to preserve operational security" and were a "convenient way of referring to subordinate and successive operations." It appears the practice grew in the Interwar years and the Americans were using Operation Names before they even entered the war with Operation Indigo.

Churchill seems to have a big hand in laying out some 'rules' for the naming of operations. MentalFloss has them down as follows:

1. Operations in which large numbers of men may lose their lives ought not to be described by code words which imply a boastful or overconfident sentiment. . . . They ought not to be names of a frivolous character. . . Names of living people--Ministers and Commanders--should be avoided. . . .

2. ...the world is wide, and intelligent thought will readily supply an unlimited number of well-sounding names which do not suggest the character of the operation or disparage it in any way and do not enable some widow or mother to say that her son was killed in an operation called "Bunnyhug" or "Ballyhoo."

3. Proper names are good in this field. The heroes of antiquity, figures from Greek and Roman mythology, the constellations and stars, famous racehorses, names of British and American war heroes, could be used, provided they fall within the rules above.

And indeed this practice is continued down the line in the modern day. Prior to the invasion of Panama Lieutenant General Thomas Kelly and General James Lindsay had a conversation in which they discussed renaming the Operation from Blue Spoon to Just Action before settling on Just Cause.

It seems that as time went on Operation Names - at least for American Military, became a means of telling which Command of the Military is leading the Operation. " Each command of the U.S. military is given a series of two-letter prefixes. The first word of every operational name must start with one of those prefixes. For example, the U.S. Africa Command was allowed to choose between three groups of letters when naming the Libyan air campaign: JS-JZ, NS-NZ, and OA-OF. By choosing OD from the third list, they arrived at the word 'Odyssey'."

To bring it back around to the question at hand: Operation Names were first used by the German military in the last two years of World War I but have since grown, developed and taken on different meanings. I imagine different militaries have different rules behind their Operation Names but the above focuses solely on the United States' armed forces.

Sources: This MentalFloss Article and The Art of Naming Operations

Follow-up question, which may drift into the 20-year rule: modern US operations tend to be given PR-friendly names like "Enduring Freedom" and "Provide Hope" - when was the shift made from more-or-less arbitrary names (Overlord, Neptune, Linebacker) to this style?

I'm assuming you not counting things like the "Nivelle Offensive" (named after French Commander-in-Chief Nivelle) as a code word.

People have been using code names for thousands of years. One example is Sheshach is used in the bible (using an early form of cryptography) as a code name for Babel (Babylon). So I'd guess that many military planners have used code words for operations throughout history, we just don't know about them.

The first documented operation that I know that used a code word is Operation Gericht, the 1915-1916 German plan for Verdun in WWI. If that is truly the first military operation with a code name, I can't say for certain why it happened in 1916. But there are some strong hints.

Most importantly, there were more ways for communication, gathering information, and spying than in past wars. Along with previous methods like human/bird messengers, physical signaling, etc. the warring nations now added telegraphs, radio, and telephone usage by the end of the war. "At the outbreak of the war, Britain, Russia, and Germany had no code-breaking establishments." By the end of the war, all of the combating nations did.

One day after declaring war, the British General Post Office ship CS Alert cut German submarine communications cables, the Borkum-Tenerife cables. Later, other British ships cut other cables or in some cases ran patch cables back to listening stations, so they could listen in on messages. Germany responded in kind. Field / trench phones and telegraphs were similarly exposed to tapping, so they obviously did not end the use of carrier pigeons and dispatch messengers.

With these unsecured methods of communication, all nations had to start cryptography departments. The British version was called Room 40. Early in the war, in 1914, the German Magdeburg ran aground. Before the Germans could destroy it, the Russians captured it. They found the German signal code book (Signalbuch der Kaiserlichen) and the current encryption key. This code book and encryption key were passed along to the Brits. Now, even if the Germans changed their codes and encryption keys, their enemies would still know the general structure of their methods of encoding.

So while I don't know for sure, I think it's safe to assume this early blow to the German communications security, added to the general lack of security of communications, and various new methods of communications, had a lot to do with the adoption of code names for military operations.

Sources:

Tucker, Spencer (editor). World War One. "Intelligence and Counterintelligence". 2005. pp 578-580

Robinson, Robert. Electronic Warfare In WWI. [pdf]

Other related reading:

https://en.wikipedia.org/wiki/Trench_code

https://en.wikipedia.org/wiki/D-Day_Daily_Telegraph_crossword_security_alarm

The UK introduced random names for operations during World War II, which lead to the use of Rainbow codes so that project names did not leak secret information. I read a book about Operation GRANBY (the British contribution to the 1990-91 Gulf War) which said names were created randomly by a computer in the MOD's (metaphorical?) basement.

I don't have access to that book now but Google Books has a similar account. I also found a blog reporting a discussion of the issue on the BBC.

I am the world's leading authority on this subject. I say that as I am the author of the only book on the subject. (Can anyone find it? It is long out of print.)

In any case, the first use of code names for operations was by the German General Staff in WWI. Certainly by 1918, names like HEILIGER GEORG (Saint George) were used in planning documents. Other names of that time and place were HEILIGER MICHAEL, HEILIGER QUENTIN, ACHILLES, BLUCHER, CASTOR, KAISERSCHACHT, and POLLUX.

But of course no full fledged system springs into being overnight. The origin probably lies in the German staff schools where various "Cases" were considered by groups of students in competition to reach the best solution.

But we can say with certainty (when Saint George was executed in the smaller form call Georgette) in April 1918, the practice was standardized in Germany.