I'm a board member/officer for my school's student nurses' association. Like ElkPark said, we're a chapter of the NSNA and of our state's SNA. We do a lot of fun things at our school, and we really work hard to try and make our school feel like a community. Every month, we have a meeting that's open to the entire student body. We give free lunch and have a guest speaker come in to talk about something nursing-related. We offer a lot of volunteer opportunities out in the community; some of these are nursing-related, like health fairs, and some are just for fun (Habitat for Humanity, soup kitchen, etc.). We also put on events throughout the year for the students at our school to enjoy. We have movie nights (especially around the holidays, we're having a scary Halloween movie night coming up), a mentor/mentee ice cream social (sophomores, juniors, and seniors get together and share advice/tips), charity fundraisers (last year we had an event at Build-a-Bear where students came and made bears for the local children's hospital), and even a nursing olympics in the springtime (where there are fun nursing-related events, such as a three-legged restraint race and a high crawl under IV tubing). For networking, we organize a career fair every spring, we work with Kaplan and other NCLEX prep programs to get special group deals for our school's students, and our faculty advisor hosts a panel in the spring where former students come back to give advice about finding a job and adapting to life as a "real nurse."
Logistically, we have 9 different board members, plus one faculty advisor. Each board member has a specific job. I am responsible for organizing volunteer events/community service. We also have a president, vice president, secretary, treasurer (manages $), image of nursing (visits high schools to talk about nursing, visits pre-nursing classes), fundraising director (in charge of merchandise - designs and sells t-shirts, hoodies, water bottles, etc.), membership director (encourages students to join the national and state nurses' associations), and a student-faculty liaison (attends faculty meetings to represent our student body).
If you could get a few other students to work with you and a faculty member, you could probably put together your own student nurses' association. It might start slow, but eventually it will grow and become more popular. My school's SNA has only been around for 4 or 5 years. I remember when I first started my nursing program (fall 2011), hardly anyone went to the meetings. As a student, I didn't even know half the things our SNA did. However, in the past year, students at our school have become a lot more interested in the SNA. At our last meeting (the first of the school year), we had nearly 90 students in attendance! (What I'm trying to say in this paragraph is don't feel discouraged if students aren't interested in your group at first. As they begin to see all the things you're doing, they'll want to be a part of it and your organization will grow.)
You should also check out the NSNA's website. They would probably have information about how to start a chapter at your own school.