Humans are predominantly right-handed because of the position of the heart: left-handedness carries an increased risk of cardiovascular disease.
The heart was tilted to the left of the sternum in bipedal primates.

The human right hand is controlled by the left cerebral hemisphere and by the right cerebellar hemisphere, which bilateral control means that in the event of a lateral head injury, the right hand can still be moved.
The cerebellum (little brain) evolved before the cerebrum (brain).

The bilateral control of the upper facial muscles means that in the event of a lateral head injury, both eyes can still be closed.