Hi Tyke, when I needed to change the name of a horse I would ask him, then go to a blank mind and a name would come up.  I'd check with the horse to make sure he'd agree.  I've done this both with changing a name of a horse and for coming up with a name for a horse who came to me without a name.

Now your only problem is if they both insist on keeping the name Bud.  Then you could negotiate a compromise.