When making a collar, the "random" button gives name suggestions. If you want to use a custom list instead of the standard suggestions (e.g. for authentic RPs, inside jokes, ...), just do this:

1) Find the mod in your game files. You can open the game files from Steam Library and then go to "Don't Starve Together -> mods -> workshop-1101091998"
2) There should be a file called "modinfo.lua". Open that with whatever text editor you want and copy the list.
3) In the same folder as modinfo, create a new file called "customnames.lua". Make sure it's an actual Lua file and not a .txt file.
4) Add your names of choice to the list and you're done!

Here's an example for the customnames.lua