I
I would recommend trying to balance the physical risks of sport with the mental benefits. I’d probably suggest going for lower-impact sports like soccer, basketball, gymnastics, or roller derby, since they still give you that team environment without as much risk of head injuries as sports like American football or rugby.
Whatever you choose, I would recommend warming up properly, stretching after, and not overtraining. I’d also suggest seeing a doctor regularly (ideally a sports doctor) who could help prevent injuries, like showing you how to support your ankles or knees. If you were doing a high-impact sport, I would recommend getting regular check-ups for your brain as well.
I would also recommend group sports, since they tend to have the biggest social and mental health benefits. Even something more "solitary" like running can be done in a group, which makes it more fun and gives you a shared goal (like a specific race in a specific time) and something to celebrate after.
II
To find the right fit, I would ask what the patient enjoys and what they’re good at. Different sports suit different people: basketball requires coordination, tennis a different kind of coordination, and skating more balance. I would ask whether they would prefer something more straightforward like running, or something more creative like gymnastics, ballet, or dance.
I would also ask what they’re looking for, whether it’s the social side of a team, the excitement of playing and scoring, or more flexibility in terms of schedule (when they want to practice and for how long). The mental health benefits of sport often come from both the social environment (especially the right one) and from endorphins. So, I would try to recommend something that best matches the patient’s medical needs and personal preferences, based on the aformentioned questions.