Oh, there was a time when bartenders had something resembling a system. If you ordered a Toddy, Smash, Sling, Sour, or Julep from a bartender in the 1890's, everything else became a modifier: I'll take a hot whiskey sling. Although, even then, creative and inventive names were part of the bartender's repertoire. So how do bartenders come up with cocktail names like the Corpse Reviver #2 and Negroni?
Here are a few guidelines when naming cocktails. I'll pick an attention-grabbing category to get you started.
Like naming horses, the pedigree of a drink matters. There are no more fun examples than Sex on the Beach variants. By the name alone you can ascertain the combination of ingredients and determine the cocktail from which it originates. There's the Sloe Comfortable Screw on the Beach, which is a Sex on the Beach with Sloe gin and orange juice (thus the "Sloe" and "Screw," as in Screwdriver). Then there is also the Sloe Comfortable Screw Up Against a Wall, which includes Galliano (the "Wall" is short for Harvey Wallbanger that uses Galliano). You can take it from there. Here are some more name sources.
Perhaps one of the most famous old cocktails--which is still a staple in bars today--is the Manhattan. Purportedly named after the Manhattan Club in New York, it also has the interesting distinction of being named after an Indian word Manahactaniek, meaning "The island where we drank liquor," according to social historian Iain Gatley in Drink: A Cultural History of Alcohol. At least Gatley claims the island is named such--the drink came much later, leading to an odd recursive effect when we drank that drink named after the place named after the place we drank.
Famous people and larger-than-life personalities often invent their own drink and pass the word around: Count Camillo Negroni and Colonel Joe Rickey are two perfect examples with the Negroni and Rickey. The Colonel went to various bars during his travels and instructed bartenders how to make his drink.
Then there are people honored with a drink name, such as with the Robert Burns and the Montgomery. Hemingway named his variation on the classic Martini the Montgomery since the gin to vermouth ratio is 15:1 (the odds that Field Marshal Montgomery purportedly favored when going in to battle).
Sometimes the liquor is essential to the name, such as the Bacardi cocktail (in fact it's illegal to state that you're serving a Bacardi Cocktail and not use Bacardi). Dubonnet Cocktail is another. These seem to lack inspiration but get right to the point. Whenever I'm stuck on a name, this makes a good placeholder: "You know, that Dolin Vermouth Cocktail."
Plays, movies, and popular songs are sometimes drafted for name recognition. The Rob Roy was named after a famous operetta, based on a famous novel, based on a famous poem about a Scottish, Robin Hood-like character. I myself enjoy using albums titles or famous songs from bands I like. I have both a Paul's Boutique and Lebanese Blond cocktail named after the Beastie Boys' best album (go ahead, try and argue otherwise) and the Thievery Corporation's breakout song.