For dogs, Morzsi, Pamacs, Blöki, Foltos are fairly common. In case you have a kitty, I'd say most of cat names are somewhat related to the sound cat's make. Such as Miau, Nyau, Miu. Since the Hungarian word for kitten is "cica", people often just play around with it and name their pets Cirmi, Cicus, Cicu, Cicamica, Micu. 

Oh, and there's a common name for cows too, it's Mariska/Riska. :D