I've noticed that my son generally experiments fine motor skills initially using his left-hand (stacking, eating with utensils, coloring) and then shifts to using his right more often. I'm honestly not sure if there's a hand he prefers or if he just goes with the one that's closer to the object to be manipulated. When does a child typically settle on a dominant hand? Additionally his grandfather (mother's father) is ambidextrous, is that a genetic trait that's often passed along?
My understanding is that the handedness is a mix of what they learn by observing their parents and other people, and of how their brain is wired and/or developed.
While my son was between ½ and 1 year old, he would have one preferred hand for weeks, and then switch to the other hand for weeks. We were guessing whether he'd eventually "land on" right- or left-handedness, but I saw nothing that could be taken as an early indicator of his final choice. He's been generally right-handed for a long time now (now 2 years old) so I think it'll stay that way.
The following seems to be scientifically disproven. I'll leave it in here for the sake of completeness:
Most people have a dominant brain half. There's a statistical correlation (meaning it's often but not always the case) regarding the dominant brain haft, with a dominant left brain half being correlated to analytical skills and also to right-handedness -- versus creative/artistic skills and left-handedness with a dominant right brain half. I'd like to quote some scientific studies here, but perhaps I can fill that in later -- I can't say whether this dominance and handedness is learned, determined by genes, or determined by other factors.