The first test, Trinity, was given its name by J. Robert Oppenheimer, for reasons that were lost to history but may have had something to do with the poetry of John Donne and the suicide (?) of his mistress Jean Tatlock.

The second test, Crossroads, was given its name by Admiral Blandy of the US Navy, chosen "merely for brevity in dispatches and communications, and we chose it with an eye to its possible significance."

The names post-Crossroads were all developed by a much less idiosyncratic bureaucracy, the Atomic Energy Commission, and things were a little less up in the air. They start to look like regular military code names, which were chosen early on, picked to be mostly purposefully meaningless, checked to make sure they didn't conflict with other code names, and then just used as a piece of jargon.

The hyphenated operations are actually two operations that were combined into one. So Buster and Jangle were different phases, but as a whole you can consider them one operation.

Chuck Hansen, in his Swords of Armageddon, has this to say on the names of the shots themselves, which varied a lot:

Starting in 1953, to avoid even more confusion in the future, shots were given unique names that were not based on the international phonetic alphabet.

Names later assigned to U.S. nuclear tests in the Pacific and Nevada were rather eclectic and reflected, at least for LASL, a distinct regional geographical influence. Shot names were generally never identical to or indicative of the devices tested and were usually carefully chosen to obscure any purpose of the test.

A notable exception to this rule occurred in Operation TEAPOT during the spring of 1955, when several shot names — Wasp, Bee, Hornet, Moth, and Wasp Prime — were identical to the classified names of the devices being fired during those shots. In addition, during Operation UPSHOT-KNOTHOLE in 1953, the shot names were first-letter mnemonics for the classified device names (an unusual security abberation).

Los Alamos and Livermore usually used separate naming conventions; however, during Operations REDWING and HARDTACK Phase I, both labs employed a common basis for shot names. Shot names were typically assigned after the selection of a specific naming category (or categories, in some cases); a list of candidate shot names was drawn up by each laboratory based on the category (or categories). This list was then submitted to the AEC's Division of Military Application which checked to determine if any of the code names had been previously assigned to other classified projects or weapons. If they were not already in use, then the names could be used in the test series.

For dual-purpose (effects and weapons development) test series names, such as BUSTER-JANGLE, TUMBLER-SNAPPER, and UPSHOT-KNOTHOLE, the names of the series referred to both AEC technological development shots and DOD military effects tests. [...]

U.S. underground tests series between 1963 and 1992 featured test shots named after wines, cheeses, vintage automobiles, tools, nautical terminology, games, New Mexico towns and Indian pueblos, historically important California and Nevada towns, golfing terms, mixed drinks, Spanish bird names, pastas, and many other things.

During the summer of 1988, the Department of Energy offered the following explanation for the assignment of test names:

From the beginning of the Atomic Age (sic), starting with Trinity — the world's first atomic test in 1945 — names have been assigned to nuclear tests. Wartime secrecy made this necessary for military purposes. Both the nuclear weapon or device, whichever is being tested, as well as anything that is remotely descriptive of it, are classified. As a result, scientists and test planners have assigned code words and nicknames to all tests.

Early tests used the military phonetic alphabet, like Able, Baker, Charlie, etc. With more and more tests taking place, more imaginative names were used. These included names of rivers, mountains, famous scientists, fish, birds, and so forth.

I will only add to above that the first two tests of the Lawrence Livermore lab were named after the godparents of the lab director, Herb York: RUTH and RAY. Which gives an indication of how idiosyncratic the process could be!