Since it is prom night I think the best thing to do is keep the speech short and sweet with a bit of humor. This is going to be a very memorable night for people and no one is going to want to listen to a long and boring speech. Highlights of the year should be included to rekindle fond memories of the school year.
People remember humor. Keep it light and relatively brief - ten minutes or less. People enjoy hearing stories about the year or the class' time in school over the years, but remember to keep it school appropriate. You might also want to tell your cousin to be magnanimous - that is, show some humility and talk about the other candidates in the class. If he wants, maybe even give the crown to someone not even in the running, say someone with physical or mental challenges. He'll be the most popular Prom King ever not to mention it's just a classy thing to do.
In terms of any closing remarks at a prom, most of them should involve gratitude and a brief note about for many of the seniors, this will be one of the last moments they are together. Graduation and Prom are usually the last time Seniors will be able to be in the company of one another. I think it would be appropriate to briefly and succinctly note this in a public setting. At the same time, expressing gratitude to the many people who actually spend time organizing a Prom would be highly effective. I would also add a note that a good time does not involve reckless behavior post- Prom, so something witty could be added here. In all, I would emphasize gratitude and a note of togetherness.
What your cousin should say depends so much upon what the tradition is at his school -- what closing remarks are usually given.
I would think that the most usual kind of closing remarks for a prom would center around the themes of the prom. Typically, a prom has a theme and your cousin could try to talk about how the prom lived up to or emphasized this theme.
I would think that your cousin would also want to thank whoever had a particularly big role in creating the prom. Whoever was in charge of decorations or something like that. He might also want to thank people like the chaperones.
Above all, I would think your cousin should keep it short...
If your cousin keeps it short and simple, they will be far more memorable than to discuss esoteric subjects. If you spend time too much time expressing gratitude, like an Oscar speech, it can be too dull, and listless. Try speaking from your heart, and tell people what you experienced so that others can share in those experiences.