Depends what your barn name is :) As far as I know, it doesn't REALLY make that much of a difference, but at higher level competitions, it's practically recquired. Although it's not a recquirement, I would definitely reccomend having a show name. I remember when I first started showing, I was on a lesson horse, and all the other riders before me were being introduced like "This is 'so and so' on Lasting Impression" and "this is 'so and so' on The Whole Sha-bang" and then it was "This is Maddy, on Popeye" and it really didn't help my confidence!
Now, I even changed my horse's show name because her barn name was better than her show name: for some reason, her stable name is "Concept" and her show name was "Holiday", but now it's "Conceptual Image", still dorky, but it makes ME feel better at shows!