Frequently it depends on the strain or type of breeding the horse has. The Egyptians are more likely to name a horse Ibn or Bint something (Ibn Al Muktar or Bint Maysa). Ibn means son of and Bint means daughter of. Generally for boys they'll name them Ibn Stallion and the fillies are Bint Mare.
The Polish tend to name for the first letter in the dam's name, Forta, Fortalla, Farmacja and so on. Russians do the same since they use a lot of Polish horses but if the foal is born in Russia it becomes a Russian Arab rather than a Polish Arab. Confused beyond hope yet?
Then there are fad names like Psyche. We have Padrons Psyche, Magnum Psyche, Pstrategy, Psitadel, etc etc to show they are all Psyche related.
Or they can show the stud where they were born, Gazal, Gazal al Shaqab, Marwan al Shaqab, then Marwan al Magnificoo because Marwan al Shaqab is the sire and the foal was out of an Echo Magnificoo daughter.
My stallion svs Il Divo crossed on my Magic Dream daughter might end up with a filly like Dream Diva DCA. SVS is the intials of his breeder's last name, DCA stands for Dreamcatcher Arabians.
The straight Egyptian names can be pretty tricky to get around but usually you can sound them out pretty close. Personal choice? I refuse to name any foal an Arabic name because of 9/11.
My stallion's name is Italian (why? I don't know, sounded good at the time I guess, his sire's name is Fornaio.) so I might name a colt Ducati DCA for an Italian motorcycle. I have a saddlebred colt named Hillbilly Harley so that would carry the motorcycle theme a little farther. There are some horses named after designers and perfumes (Escada Bey, Stetson Bey, Cacharel, etc.).
Some folks take the naming of a foal SUPER seriously and you can offend them with a funny name. I can't tell you the number of breeders who gave me pure Hayell for naming Harley, Hillbilly Harley DCA. Oh well, I bred him, my horse, my name. Besides, he IS a bad little motor scooter. Someone I know tried to name a palomino half Arab Acapulco Gold and their friends hit the roof.