Lots of good advice here thus far.
My dad had me with him hunting and fishing since I was old enough to walk. I would just tag along... squirrel hunting, scouting for deer, tracking and field dressing deer. It wasn't until I was 8 years old that I got my hunting license. Dad started me out on small game... only hunting rabbits and squirrels. It wasn't until I was 12 years old that he let me go deer hunting. I know that every child is different, so the appropriate age to allow a child to go hunting will be different. But IMO, it takes an entirely different level of maturity and patience to go deer hunting than it does to go small game hunting. Personally, I don't think I would start my kid off in the deer woods. I'd take them small game hunting for 2 or 3 years before considering the next step.
One more thing... and this is simply my opinion... I'm not passing judgement on anyone else or anyone else's kids. IMO, if a kid can't go deer hunting without taking a laptop or a gameboy or an ipad or whatever to keep them entertained, maybe they shouldn't be going. I mean, if they can't sit there an enjoy the woods for what it is, are they really going for the right reasons? I just look at it like, if they're sitting there playing games the whole time, paying no attention to their surroundings, they're simply going along as a shooter, not as a hunter. "Hey, put that thing down... Here comes a deer." -- "Huh, what? Oh ok... BANG... Wow hunting is fun!" Don't get me wrong, I've played my share of Yahtzee while sitting on stand for an all-day sit.... But I don't really agree with using games and distractions as a way to keep them interested in hunting. Like I said, this all only my opinion. Keep in mind, I don't have any kids of my own and this could all change... but it's how I feel as of right now.