The thinking process is also called "System Engineering" as an engineering field of study. Beach City Robotics team 294 has a excellent presentation on it for FRC robotics
http://www.bcrobotics.org/wp-content...ering%2025.pdf I stole from it for presenting a design process for my team.
As for making the robot yours instead of the mentor's, the easiest way to do that is to get ahead of the mentors. Prepare the kids ahead of time for how to do the decision making, make plans for the kick-off and be ready. Keep the adults in the loop, you don't want to freeze them out, but let them worry about other things (money, permission slips, etc). There is always more work that could be done than the team can get done. Your team is striving to find a process for building a robot, you can help that along by getting involved in it.
That said new mentors and sponsor are prone to over control the robot build, they want good results, and there is a tendency to view the robot as the result of the team. That is not true, good students are the desired results of the team, the robot is a means to get them. By leading you leading the students in the process of building a robot, learning a little bit of all the fields of engineering involved in making a robot. You are showing them the results that the team wants to accomplish.