Often musicians comment that their physician recommends squeezing a small ball to increase their hand strength. I find this very interesting. If the individual has hand weakness from either carpal tunnel syndrome or from being immobilized (in a cast for example) this might be an appropriate exercise. Otherwise, I don't understand the need for such an exercise. Playing an instrument employs tremendous hand dexterity and muscular endurance - producing a strong hand and fingers. If a musician has recent pain in the hand, due from some sort of overuse, or "mal-use" and has been playing for years, I doubt the hand muscles are weak and ball squeezing exercises might merely prolong the pain or continue the aggravation.
Now, there are strengthening exercises that most every musician would benefit from. Number one is to have a strong core (trunk strength) to assist in prolonged sitting or standing postures, helping to support the spine and ultimately the extremities. I highly recommend fitness center core classes or "Pilates" classes. Number two is to strengthen the shoulder blade (scapula) muscles called the trapezius muscles. These muscles move and support the scapula - and position the shoulder for musical performance. They are often quite weak (from disuse and poor posture) causing shoulder fatigue and painful tension. If the shoulder is not being well supported by these muscles other muscles in the arm (not intended for this) are called into action - these muscles are often the forearm muscles which try to help support the arm as well as move the fingers and wrist for performance. They quickly fatigue, become inflamed and tendonitis arises. So keeping the the trapezius muscles strong can help to prevent arm tendonitis!!! The specific exercises for these muscles can not be described in this short blog - a physical therapist or trainer knows how to teach these exercises. (be careful, there is a muscle next to the trapezius muscles called the rhomboids which most likely does not need strengthening and can be detrimental to shoulder support and action - make sure your trainer knows the difference when prescribing exercises).
And thirdly, cardio vascular strengthening is extremely important. Every musician should perform a cardio workout 2-3 times per week. Again, see your therapist or trainer for specific instructions (or join an aerobics class).