Presumably Four was given the same choice when he first arrived in the Dauntless home. Yet, how did he know what name to give himself? His name was
based on the Four fears he encountered during his testing
Yet, it would be remarkable that he would have known exactly what those would be before he completed it. In fact, he wouldn't have known anything about the process. Too, while many people might know things about themselves, I think it would be odd for him to pick them so perfectly. Or, even if he knew what they were, for him to know what would be coming and that there would be a meaning to the term "Four".
I have only seen the movie. Does the book provide an in-universe answer for this? If not, is there a canon answer which might explain how this could have come about?