What exercises are for patients who lost some function in their hand due to a stroke?

Functional. The best exercises I know for someone who still has some partial function in the hand after stroke are called functional exercises. This means that rather than doing a boring repetitious exercise, the brain will respond better if you do something that is meaningful like trying to use a fork to feed yourself or trying to use your fingers to button a button or even trying to write.
Several. Two excellent exercise for the hands include gently squeezing a soft rubber ball (flexion of the fingers) and bringing the tips of the fingers together, wrapping all five digits with thick rubber band, and stretching the rubber band by opening the fingers. This should be done several times a day, each time doing one to two sets of twelve repetitions.