It would be nice to have more stat tracker machines that were not so expensive, like the exit velocity of the ball. For volleyball more radar technology that would help with arm velocity. We have some of these things right now but the cost is so expensive that we cannot afford them.

The students at my school love Badminton. Badminton is the most popular content area I offer as an elective. My department holds a senior class badminton tournament at the end of the semester, student compete in a an entire day school wide double tournament. As much as I feel my students are very skilled at the skills and the game, there still is room for improvement on skills and technique. I think if there was a speedometer, or speed/distance tracking device inside the shuttle it would be very helpful for students to understand the skills. Skills such as the smash and drop shot are a very difficult skills to master and understanding the height of the birdie in the air and the speed of birdie will help them make an educated guess as to the type of skill that would be most appropriate or successful to return to the opponent. The tracking information would be read on an iPad through an application. This device would be most beneficial when student are learning individual skills. This idea would also make the learning experience more interesting.

The ideal piece of technology I would like would be a heart rate monitoring system that is able to track all students from within 500 feet. The student would wear a small non-obtrusive device that sends their heart rate to a central computer that would be able to keep a log of their heart rate and calculate averages. The device the student is wearing would also be able to show them what their current heart rate is at any given moment. This would allow a student to self-assess their effort and be able to use the data to make fitness decisions. Such devices would make it possible for the teacher to see if a student is reaching cardiovascular fitness goals since heart rate and exertion vary from student to student. A teacher would be able to give breaks to students at optimal times and encourage/push students to work harder when necessary. It would also be helpful to monitor children who have a medical conditions such as tachycardia.

There are blue tooth heart rate monitors currently but the range is not 500 feet. The range is closer to 200 feet outside. Inside it depends on how many cement walls you have between the HRM and the computer/tablet. I would like to see the range extended as students can be further away especially outside. We use the Polar H7 and the Polar App (the free version is Polar Beat or you can pay the yearly fee to use the GoFitApp which stores all recorded data and is accessible on line).

An idea that I think would be really cool to have as a coach is to have each player with chip/marker on them during games with a computer/screen available. You could "track" each one so you could give real time strategy on positioning during timeouts/halftime. This would allow student-athletes to actually see where they were and where they perhaps could have been to make their team more successful- this would be great in sports such as field hockey, soccer, and lacrosse. I know video analysis is now available but for some of us that is at least a day later. I also think along with that a tracker for the ball cold also be fun to try out. Angles and speed of shots would be especially helpful for goalies.

I like this question because I'm not alone apparently. I often have these types of thoughts in the effort to get realtime data or more accurate data both in PE class and coaching baseball. The funny thing is, we are getting closer to the ideas I've had over the years.
The 2 obvious ideas that are actually happening are students who have a realtime record (visible on a wall, phone or watch)of their heart rate, calories burned, distance covered, speed, etc... and it all be available in a digital folder for evaluation and review by themselves, teachers, and parents. This can remove a significant amount of subjective data and reduce "human error". In the same idea, I wanted a strike zone in baseball that says it was either a strike or a ball and not up to the umpires who are all different in their own way. This would bring consistency to the zone and the pitcher's expectations.
To piggy back your idea of the shot put, I'd like all of the sports to be able to track the ball's angle, velocity, distance, rotation etc,. No more arguements if the ball did or did not go in the goal, over the goal line, over the fence etc.

Virtual reality in the form of google cardboard came out this past year. One thing that would be really cool would be to allow students the experience of attending sporting events they have never been to and to be a participant. For example, one of my students told me he has never been to a hockey game before. It would be great if he could have a virtual reality experience where he attends the game, but is a participant and has all the skills the hockey players have to us. So not only do they get a first-time experience, they also get a work-out.