The arc of Moises Kaufman's The Laramie Project is the gradual reveal of the need for tolerance and acceptance toward homosexuals wherever they may be, even in small towns like Laramie. The arc also revolves around the need for the Tectonic Theatre actors/performers (as surrogates for the audience) to understand the attitudes and behaviors that led to the murder of Matthew Shepard. The play is less plot-centric than it is a character study of an entire town; as a result, the arc is about how the town reacts to this murder. First, it reacts with disbelief, as they could not believe that something like that could happen; second, they start to examine their own attitudes and the way they treat other homosexuals in town; and third, they see the culture outside town (Fred Phelps, etc.) that leads to these hurtful attitudes and send a message by convicting Arthur Henderson to two life consecutive life sentences.
The character arc of Romaine Patterson is not extreme or particularly dramatic, but it is nonetheless effective, as she remains a proud and charismatic lesbian who is not ashamed of who she is or who Matthew is the entire time. Her stories about Matthew are more or less always positive, as she describes just how gleeful he was and how much of a future he had. The end of the play, in particular, shows Romaine's unique dedication to him and his cause, as she leads an angel action to block out Rev. Fred Phelp's protest of Matthew's funeral. This ultimate action demonstrates her transformation into a gay rights activist, whereas before she seemed to simply deal with her own personal life.
References
Kaufman, M. (2000). The Laramie project.