First of all, I would like to start by saying, there is no one right pet. A writer can choose to have any pet or pets they want. However, there are certain animals with lifestyles that fit easily in a lifestyle of a writer. So let's see what some are.
Fish - sounds simple enough. Due to a writer's relatively busy schedule a fish is a great companion. They don't require a lot of attention, looking at them are a great way to de-stress, and if you need to travel on a moments notice, you can easily feed them a vacation feeder. Some great, easy to take care of fish are betas, goldfish, fancy guppies, or bloodfin Tetras.
Cat - cats are pretty independent creatures; they groom themselves, bury their feces in a litter box, and often sleep through the day. If, as a writer, you are looking for something that will give you attention but also allow you to work, a cat might be the perfect option.
Amphibian - turtle, frogs, and salamanders also make great pets due to their docile nature and lack of constant attention.
A mouse, hamster, or guinea pig are also relatively silent and do not demand a lot of upkeep.
Here are some pets that may or may not be the best option for a writer:
A dog - dogs are cute and cuddly, but they can often make noise, need lots of stimulation, and move around a lot. For these reasons, a dog might be a distracting pet for writers who really need to focus and get into the zone.
Bird - while some birds are silent; others are not. A constantly chirping bird may also be a distraction for a writer. Their also constant moving and rattling around a cage could also be bothersome to some.
Farm animals - While having horses, pigs, and goats is not a bad thing, these large animals usually demand a lot of time. From feeding, training, and cleaning up after, writers may find themselves unable to write as much or frequently.
All in all, there is no wrong or right animal. It depends on the individual. All pets can help with a writer's emotional well-being and prove to be a great companion. However, if you are a new writer and don't have any pets, it may be wise to consider the pros and cons of each type of pet.