If the horse is registered then I am not sure you can change it's name?? Certainly can't over here.
If you are just talking about the regular passports then sorry I can't help you as I have no idea whether you can change them or not. Why does it matter what is on the passport though - if you don't like the horse's name then just change it but leave the passport with the not so nice name.
Don't know about changing on passport but my horse competes under a different name to that on his passport, cost me nothing just have a note on passport saying horse competes as.... signed by vet. Not sure if this helps or not!
It would have cost me £18 when I got Daisy's passport changed into my name when I bought her but her original name would have still been after it in brackets so I couldn't see much point. Her original name was Kalla so don't believe people that say you have to change it to something that sounds the same, she didn't mind in the least
She is known as Daisy at the yard but will be competing under yet another name next year