There is really no formal process for naming a train, especially a local, it is not an "official" name in most cases. Premium service such as passenger, intermodal or perishable will have a name designed by the marketing department or some group as a marketing tool. But most freight trains it is purely a nickname and doesn't appear anywhere official on any documents. For example, my description of the L329-L330 being called the 'Salty". Everybody knew which train you meant when you said the Salty, but the name was not used on any official document anywhere on the railroad. It didn't appear on transportation plans, on train lists, on track lists, on schedules, nothing.
As an example of how these names originated, I was asked to submit a suggestion for a name for a new intermodal service (prototype railroad) that ran over three railroads from Georgia to Los Angles and back. Those premium services are normally named after things that are considered "fast". I came up with Los Angeles (LA) to Southeast (SE) rail (R) service or LASER service.
Some names are descriptive, and some are ironic, and some aren't very flattering. There was a train from North Platte to Houston that carried a small block of trailer traffic, the NPHOT. Since it was basically a freight train, it acquired the VERY unofficial nickname of the "Not Particularly Hot" train.
Most trains have a very pedestrian name, usually where the train goes or what it does. A local that runs from Durand, KS to Fredonia would be the Fredonia local. A unit wood chip train to a paper mill might be the chip train. Sometimes that get a cool nickname like the "Termite train", others just stay the boring "Chip Train" for as long as they run. Most of the cool or humorous nicknames are bestowed by the crews and are, once again, not official in any way.