What type of show will you be going in? If it's a schooling show or unrated show, then it doesn't really matter what you call your horse. If his name is Bob, you can show him as Bob. Or you can make up a show name.
If it's a breed show, your horse will use his registered name.
If the show requires you to join a club or association in order to compete, then they'll make you pick a name for your horse (it could still be "Bob" if you wanted to and if no one else has claimed that name already) and then your horse will have to use that name for all of their shows.
I'm guessing the show you'll be doing is pretty casual so you can call your horse whatever you want!
I compete in my state's hunter/jumper association. Both members and non members can participate, and this year I was not a member, so I had fun with it and called my horse something different at each show ;) Next year, though, I'll pay the membership fee and compete for points since I feel my horse is ready for it now, so I've come up with the name I want to use and he'll have to go by that name all of next year (he has a registered name too, but since we are not competing at AQHA breed shows, I don't have to use it, which is good because his registered name is ugly!)