When I leased, I'd always make up cute show names for the horses, even if I knew their registered names. Vin became "Vini Vidi Vici", Woody became "Woodwind Melody", Sage became "Sage Pass Matine" Since I wasn't going for any awards or anything it really didn't matter what I called them. My trainer's daughter has shown her pony, Rainy, as Rainy Superpony.
It also sounds more impressive to hear "Amy Smith riding Red Hot Poker" being announced over the loudspeaker instead of "Amy Smith riding Pokey"
Most people will show under the horse's registered name, or for unregistered horses just make up something like I did with the leased horses- they may or may not always use the same name, but it makes it easier to find records if you need to prove your scores later on (e.g. if you were selling the horse) if you use the same name each time.