Ideas for a High School Math Club
I would definitely go to any math and engineering competitions nearby. Put together a team and compete in various competitions. Try to hold your own competitions: put out various problem sets for people to solve with prizes, etc. Find interesting problems and explore them.
Try to invite speakers on various topics from local higher education, and mathematicians/engineers in industry.
See if there are any local organizations that promote STEM education...they may be able to help in many ways. Many in the profession would love to help, I am sure.