I don't see how people got so attached to their dog that they felt sad when he/she dies. I heard that Lionhead wanted that to happen and they wanted the Dog to be one of the most important things in the game...which to me ruins what Fable really is.
Anyway my dogs name...was...umm...why would I remember?
You could always be the typical Fable Player who just wants to have fun and name him an obsene name like the Villagers. All of my Villagers are named Red Rocket, Britney Spears, and Billy Bob Jo. Couldn't hurt to name your dog like Poop Face or something. Just one of those random out of the box sort of deals.