I have had the experience of having my brain bounce in my head. I agree with some of what people have said here including that it may not be the brain, and that we cannot directly connect the pain / headaches to the bouncing feeling as different people could have different problems. Also it is very important to try to not "think" about it and focus on it, as that make the problem worse. Although I do feel that some people, weather sensitive to your own body or not, have internal functional systems that with most people happen automatically but for what ever reason some of us have to consciously work to make our bodies function properly. The good news is that after you "re-train" your body it will eventually do it automatically for you.
The only success I have found in preventing the feeling of my brain bouncing in my head is consciously strengthening and flexing the muscles in the back of my neck. It is possible that it is a skeletal muscular issue where most people naturally have their internal organs held in place by their muscle tension. I find that if my back neck muscles of flexed they shake a bit. This means that they aren't very strong. But when I flex them really hard the bouncing feeling happens less. When I do neck exercises and weight lifting it helps even more, but since I've been doing less weight lifting the problem was getting worse.
Hopefully this tip helps some people, but it can take time to strengthen neck muscles and focus on keeping them flexed when jumping. It may help some people, but it is not going to deal with all of the problems people have mentioned here.