Summarize:

Throwing and catching a ball with your left hand is a fun way to strengthen your left hand while also improving your hand-eye coordination. Simply squeezing the ball firmly in your hand will also help to strengthen the fingers. Playing tennis, squash or badminton while holding the racket in your left hand is a great way to strengthen the hand, which will give you greater control when writing. Use a small 5-pound (or less) weight and lift it with your left hand. You can also try to exercise each finger individually by lifting a very small weight with each finger of your left hand. Switch the controls on your mouse if you want, but you can still use your mouse with your left hand with the default controls. Also, try pressing the spacebar with your left hand. It's harder than you think!
Practice throwing a ball. Play racket games. Lift weights. Use your left hand to operate the controls on your computer.