Well, that depends on why you want an immortal dog. :) You could pick a dog that's sentimental to you, such as your first dog ever or the first pup you bred. Or perhaps a dog that was a gift to you, or maybe a dog that's especially beautiful. Or, you could have an immortal dog for a practical reason, such as breeding for a very long period of time, or picking an old dog that you aren't ready to have die yet.