My team actually hit this wall for the first time this year, and I think we handled it pretty well. Let me detail this out for you.
General Interest Meeting: Students came into a theater or whatever, and we did our basic presentation about the team and why they would like to join. At the end of the presentation, we told the student to fill out an application and to show up at a “tryout” day.
Tryout Day: We gave the students some basic tasks and/or challenges. We were not looking for technical skills or even how well they completed the challenge. But rather, we were looking for people who worked well in groups, showed good logic in their thought process, and ultimately looked like they would be a good fit in our environment.
After the tryout day, student leaders and the head mentor met up to select the applicants. They chose the students based off what they saw at tryouts and applications.
Note about the application, we asked general, census information (name, age, contact information), but we also asked students to write about the time they worked on a team, why they wanted to be on our program, and some stressful situations they have had to deal with. The application didn’t put an emphasis on technical skill or prior robotics experience, but students who were involved with FLL and such could brag about that.
During the process, we were honest with the students about our expectations and our process. We emphasized that we were only taking 16-20 students, prior robotics involvement was not a prerequisite, and that we were looking for folks who would mesh well with our team.
In our mind, we didn’t set a very high bar for applicants, but asking them to jump over a bar let us see how high some folks could really go. As such, it was a good process, it was remarkably fair, the students and parents accepted it, and we will probably do something very similar in the future.