In the series, this was the first title that mentions a person rather than an object. This title also stands out for being the most misleading in the series thus far... which happens to be really fitting. Azkaban is largely a mystery story (with elements of fantasy and Bildungsroman – a fancy German word for a coming-of-age story). The title helps set up the mystery elements of the novel from the get go – who is this prisoner? What did he do, and what does he want now?
Of course, just when we, and Harry, think we have the whole thing figured out, the novel throws us for a loop and we learn that we were wrong about everything. The Prisoner of Azkaban (as in, the character, not the title) isn't the real villain of the story at all. Consider our minds blown.
This title also ties in two more of the book's major themes – family and the past. As we learn more about the prisoner, one Sirius Black, we start to get an entire story-within-the-story – a history of the conflict with Voldemort, a crash course in the lives of James and Lilly Potter, and the strong family ties that Harry, one of the most famous orphan characters of all time, still has in the wizarding world. The Prisoner of Azkaban starts to work as a kind of metonym, which is a fancy way of saying a word or short phrase that stands for an entire concept or idea (try busting that one out on your next AP exam!). The Prisoner here stands for the entire first war with Voldemort and the ways in which Harry's personal history is closely linked to that conflict.