Who is the better person in Mary Shelley's novel Frankenstein: the monster or Doctor Frankenstein?

Because this is a subjective question, it is certainly up to each individual to decide based on his definition of "better." Establishing some criterion for who is the better human being will help you make your own decision.

First, determine which of the two characters most values life. Victor Frankenstein seems to be the one who most values life, since he manages to give life to another human being; however, as soon as it is clear that the monster does not meet his expectations, Frankenstein promptly and thoroughly disowns him. On the other hand, the monster wants to become part of society but of course is shunned. Though he commits several murders, he has no other outlet for his hurt than to take revenge on the man who gave him life and then disowned him. So, one man gives life but wishes he had not; another wants to life a full life but cannot and instead takes several lives to punish his careless creator. Who values life more?

Second, examine the behaviors and motives of both men. Frankenstein is driven by his selfish desire for more knowledge (and his pride in wanting to best his teachers and fellow scientists). He displays little interest in the end result, not as a man and not even as the product of his experiment. The monster, as we know, did commit several murders; however, he also did several kind things (such as saving a girl from drowning) and does feel guilty for his wrongdoings--something Frankenstein never does. Which of them acts and thinks in ways that display the better aspects of human nature?

Finally, determine which of them exhibits more human emotions and empathy. The doctor acts only in his own self-interests throughout the novel, and when the monster asks (okay, demands) that he make a female monster for him so he will not be so lonely and such an outcast, Frankenstein cruelly reneges on his promise. The monster regretfully says, “...once I falsely hoped to meet the beings who, pardoning my outward form, would love me for the excellent qualities which I was capable of unfolding.” Which of them exhibits greater humanity?

If you decide that the doctor is the better person, it is ironic that people who have not read the novel  (or seen the movie) assume that Frankenstein is the monster in the story. If you determine that the monster is the better human, it is apt that Shelley gave such a monstrous name to the title character of her novel.