Writing a persuasive paper in a Shakespeare class, like in any literature class, requires a thorough reading of the text. For example, after reading several Shakespeare plays, you could argue that textiles, like the handkerchief in "Othello" and the importance of clothing in "Cymbeline," play a crucial role in the development of characters in Shakespeare's plays. Examine the use of cloth in several plays and compare them. Convince your reader that you can understand the characters of the play based on how they relate to the textiles.
In a world history course, the available topics for argumentative essays is as broad as the class's subject matter. You could convince your readers that the effects of Alexander the Great's conquests did more to unify the world in which he lived than other conquests by European countries much later; keep in mind that the "world" was defined differently for Alexander than it was for later Europeans. Use examples of the rapid permeation of Macedonian influence into the cultures of those conquered by Alexander to demonstrate a unification of cultures across the civilized world.
If you are taking a class on language or linguistics, you could argue that certain methods of word formation are more effective than others at getting new words adopted into the language. Study the history of words that have been recently adopted into English. Use modern examples, like the use of "google" as a verb, or the word "xerox" to refer to photocopying in general. Also, "kleenex" refers to all facial tissue.
If you are taking a class on sociology, there are myriad options. You could look at the prevalence of reported depression in high school or college students, then look at changes to school curricula -- like increased standardization and pressure to pass tests -- and argue that such changes did or did not effect the increase or decrease in depression. You could use benchmarks like the passage of the No Child Left Behind act, and see if there are increases or decreases in the prevalence of depression after its implementation.