Here's what we know so far from a scientific standpoint.
There is evidence for genetic influence for handedness; however, it is non-Mendelian and geneticists cannot agree on the exact process. There is evidence that handedness can be influenced (and changed) by social and cultural mechanisms. For instance, teachers have been known to force children to switch from using their left hand to using their right hand for writing.
Some researchers argue there is evidence for cases of "pathological" left-handedness related to brain trauma during birth. And many researchers trace the cause of handedness back to pre-natal, intrauterine developmental processes, back to the time when the fetal brain is first developing distinct cerebral hemispheres. In the 1860s the French surgeon Paul Broca noted a relationship between right-handedness and left-hemispheric brain specialization for language abilities. But the hand-brain association is neither a simple, nor reliable, correlation. Studies conducted in the 1970s showed that most left-handers have the same left-hemispheric brain specialization for language typical of all humans--only a portion of left-handers have different patterns of language specialization.