Noonian, like his grandfather, Arik, was left-handed, but why would he also establish the characteristic in his android creations?
Data (and Lore) are left-handed because Soong scanned his own brain in order to provide the basis for their positronic brains. Soong himself was left-handed and his neural pathways were evidently laid down with the same handedness.
Later, Data attempts to replicate Soong's work by scanning his own brain and transferring the result to his offspring. While Lal's handedness isn't canonically confirmed (she uses both hands on multiple occasions and appears ambidextrous), it's still a pretty good baseline for the techniques used in the process:
DATA : There was a new submicron matrix-transfer technology introduced at the conference which I discovered could be used to lay down complex neural net pathways...
WESLEY : (understanding) You mean a transfer from your your brain to Lal's.
DATA : Exactly, Wesley. I realized for the very first time it was possible to continue Doctor Soong's work. My initial transfers have produced very encouraging results so I brought Lal's brain back with me to continue. - TNG : The Offspring
As @JasonC has noted in his comment below, Data appears to be capable of using both hands equally well but only does so when he is acting on a fully conscious level. He does seem to favour using his left hand for "muscle-memory" tasks like painting, playing music, using a pen or firing a weapon.
Data could use both of hands equally well, so I don't think it's accurate to call him "left-handed"--he was ambidextrous, although Memory alpha does say that he tended to prefer his left hand for various one-handed activities.
With that said, I don't believe this quirk was ever explained on the show. Some speculative explanations include:
- Maybe Data was simply programmed to default to his left hand if there was no reason to use one hand over the other. Left = 0, Right = 1 in his brain or something.
- It's possible that Soong deliberately engineered this predilection such that Data would have more of his father in him, just as a human child would.
- It's possible that Data in fact had no such preference, and we just happened to see him using his left hand more often by random chance.
It may have been a conscious decision on Data's part to exhibit handedness. It would certainly be in character for him to choose to have a dominant hand in order to seem more human. In the episode Birthright, Part I, he is shown to paint two paintings at once, one with each hand, which seems to indicate that he was in fact ambidextrous.