It is interesting that Myrtle wants a dog for several reasons.
First, dogs are known as "man's best friend," and something within Myrtle seems to stir toward loyalty. She isn't being loyal to her husband, and her lover is married himself; yet, here she is, reaching out for something to fulfill this void in her life.
It's also interesting that she seems to have no foresight into the care required by a dog. She says, "I want to get one for the apartment. They're nice to have—a dog." It's really unclear how Myrtle plans to care for this dog. After all, she is only really there to meet Tom for their little rendezvous. But, in the moment, she makes a quick decision based on an emotional tug and takes a puppy with her. Myrtle seems to be led by her emotions.
In the next scene, she antagonizes Tom about Daisy until he breaks her nose. Later, she is overcome with emotion and runs into the path of the car Daisy is driving, which kills her. As seen with the dog, Myrtle isn't led by a sense of reason. (As a side note, the question of what DID happen to that dog is one of the mysteries which is never answered in The Great Gatsby.)