In Star Trek III - The Search for Spock, we are presented with Excelsior's Captain Styles, a bit of a preening martinet. He is first seen greeting Mr. Scott in Engineering, and he is carrying...something. A riding crop? A SPAAAAACE RIDING CROP? What is this thing? We see it again when he is in his quarters.
What is it? Does anybody have any idea? If it's a riding crop, then for God's sake, why is he carrying it around on board a starship?
According to an interview with James B. Sikking (Captain Styles) on Startrek.com, we learn that it was a swagger stick, which was a military accessory common in the 19th and early 20th centuries.
Q. Whose idea was the swagger stick that Styles wielded?
SIKKING: I think… I think… I don’t know. I’m not sure if it was mine, but it was certainly a good idea. When I was studying acting years and years and years and years ago, we’d have a class and they’d say, “Go to the bus depot or to the train station or the airport. Take a little bag with you. Wear dark glasses. And go sit somewhere and observe people.” It’s true; you sit there and watch people, and how they move their bodies is far more communicative than dialogue. So any kind of a prop that reveals a character is well worth having in your hand or next to you.
Startrek.com - "Hill Street Blues' Lt. Hunter Was... Star Trek III's Captain Styles"
The goal was to visually convey the script direction about Styles' personality
The elevator door opens, and as Scott is preparing to step in, CAPTAIN STYLES steps out. Styles is an officer about Kirk's age. If he's a little stuffy, it's pardonable; he does, after all, have the plum assignment in all of Starfleet.
I think you are close with your idea of a riding crop. From his attitude and context of military command I am pretty sure it is meant to be a swagger stick. In this case of the riding crop variation.
As a martinet this would be right in character for such an officer.
