Can tai chi help to rehabilitate the aging brain and keep your brain young?

It can indeed. As with most physical exercise, tai chi can be very good for keeping both the body and brain active and, as a result, young. The attention to detail in learning the specific details of the movements is very good for brain and body and is a very low impact form of exercise. Although at its heart it is an extremely effective form of martial combat, tai chi is very useful for relaxation and meditation.
Yes. I see 2 major benefits to tai chi. One improved balance. Balance issues are huge in older people. If an elderly person falls and breaks a hip, the implications are enormous and 50% will die within the year. Improved balance keeps people active and this stimulates the brain. The other benefit is improved mental state which helps to stave off depression and dementia in my opinion.