For me it would depend on the horse. What he or she looked like, or any personality traits I observe could be a source of inspiration to me when trying to think of a nice name.
The name Sham though I think would be a pretty cool name if I had a horse. It's the name of the racehorse billed as rival to the famous Secretariat. I have always liked underdogs, and think good old Sham deserved more love. He was a noble-looking, dark, chocolatey bay horse, and would run beautifully. He was a great horse, he was just paired repeatedly against the best thoroughbred out there, and no horse in 50 years has been able to surpass the Kentucky Derby record set by Secretariat I think. So it could be a nice tribute to name a horse after the hard-working runner up, and not forget about him.
Shams is an Arabic word meaning sun, and in the children's book King of the Wind, it's the name of the Sultan of Morrocco's horse, whose changing fortunes and owners take him around the world. He's underestimated because of his smaller size, but he can ride like the wind.