Some of the central characters in "Frankenstein" are notable for their physical appearance, with destinies that may or may not be reflected in their looks. Argue one way or the other. Related to this idea, you could also argue that physical appearance is an indicator of character. Discuss whether beauty signifies a beautiful soul and whether ugliness represents an ugly soul.
You might take a position on who is worse, the monster or his creator. Consider which of them is more a monster or whether they both are monsters. You will need to look past the superficial and consider each character from different angles, the physical and the spiritual among them. You will also have to set criteria to define inner and outer beauty and use it to make a comparison of each character. Related to this, you could develop a thesis that answers whether the monster, Frankenstein, Caroline, Elizabeth and other characters are symbols of human nature.
Mary Shelley wrote "Frankenstein" as a Gothic novel during the Romantic period, during which the natural world was revered. You could write about the ways in which the monster represents the natural and the unnatural, arguing that he is more a reflection of one or the other, or that he represents both equally. In either case, discuss the moral values of Romanticism. You might also consider taking a position on whether or not we should view the monster as a victim or perpetrator in light of Mary Shelly's Romantic sensibility.
Through their letters, several characters inform the storyline of "Frankenstein." Some critics have argued that too many characters drive the storyline, making the work a mishmash. You could develop a thesis on that narrative strategy, discussing whether it works cohesively, or whether Shelley should have eliminated some of the epistolary influences, instead sticking to third-person omniscient. To do so, you will need to consider whether her strategy serves a purpose that is not readily apparent, such as reflecting the monster's creation, or whether it was just a quick way for her to develop the story without having to finesse point of view.
Alter egos often populate works of fiction, and it's always interesting to argue that two characters do or do not represent different sides of the same character. For this thesis, discuss how Frankenstein and his monster are the same and different, and in what ways they seem to mesh. Ultimately, you will have to take a position on whether they represent different aspects of what is essentially one character, or whether Shelley intended them to be taken at face value.
Think outside the parameters of the story and compare or contrast it to another work, such as "Faustus," a German tale later turned into a play by Christopher Marlowe, "The Tragical History of Doctor Faustus." Argue that Dr. Frankenstein and Faust share the same tragic flaws, or that they are not, in the end, alike. Along the same lines, it might also be interesting to compare Dr. Frankenstein and the monster to Herman Melville's Captain Ahab and the whale in "Moby Dick." Any work that offers two strong characters at odds with each other will provide fodder for a paper that also discusses "Frankenstein."