I've recently started running one-on-one games for my 4.5 year old. (I should note that waiting to age 8 or so is completely reasonable--that's the age I started at
--but that my son was pushing to start earlier, and has a really solid sense of imagination, so it made sense for us to start earlier.) I do think that for very young children, a fair amount of simplification is necessary. My son can recognize the numbers on the die, and understand that high is good and low is bad, but he can't reliably do even the simple addition of a modifier to the die roll. We're using a (slightly simplified) 4E rule set, but that's to maintain compatibility with the games that a friend of his plays with his father--maybe a third of the games have been joint games. I wouldn't recommend starting with 4E, however: the additional complexity is tough for very young kids. A fighter whose standard action in combat is just to attack can be good for kids. If it were not for wanting to let him play with his friend, I would have gone with a simpler system--probably as simple as "roll high and you succeed!"
BECMI is certainly an awesome system. For a young player, concentrating on the BE side of things will make life easier.
That said, for young players in particular, some of the old baggage of earlier editions of D&D will be problematic. There will be a point when a young child can do addition well but not subtraction. Descending armor class makes life harder. (Yes, you can convert this into a table look-up approach, but I think it's better for young children to spin it around into an ascending scale system.) For those reasons, I might consider a stripped down 3.x system or one of the retroclones that incorporates "high is always better" and "roll a d20 and add your modifier is the core mechanic," rather than pure BECMI. But again, this depends on age of child and the individual child--for a really young child, or a child for whom any arithmetic will be frustrating and unfun, I would be inclined to start with a proto-system: roll a six-sided die, 3 or higher succeeds--for really hard tasks, you need a 4 or even (gasp!) a five! For an 8 or 9 year-old who is a strong reader with strong arithmetic skills, throwing them into the deep end of 4E or 3.x or AD&D, whatever flavor is your preference, would likely be fine. If you want to wade in by starting with a simpler (or simplified) system, that would be fine, too.
Some other comments: especially with young kids, shorter sessions can be your friend. I typically run games for my son that are less than an hour--maybe half an hour for a normal game, an hour for a long session. That works well with managing his attention span. That means that the adventures tend to be pretty straightforward--maybe 2 or 3 "encounters" long, with a mix of mostly combat, some traps and puzzles, some role-play encounters, and so forth. As a player in a game for adults, I would find those frustratingly short and simplistic, but for a young child, those are just right--longer than that would be frustrating and lead to wandering attention.
Also, you should think carefully about whether to start off with just your child or whether to include one or more friends of the child as well (perhaps the children of your gaming buddies, if that makes sense for your circumstances). There are real trade-offs here. My son really enjoys playing D&D with his friend, but they are also much more out of control and prone to wandering attention when there are two of them at the table. Also, the combats last nearly twice as long, which means that in place of a quick, fun combat, sometimes you have a combat that lasts too long. For really young children, my advice would be to start them off playing on their own, and then have occasional very small group (say, 2 players + GM) games to supplement that. For older children who have more ability to focus and self-discipline, it might make sense to start with a group off the bat.
For more comments and thoughts on this, you might check out
my story hour of games with my son.
Running games for my son is one of the most special things I get to do. I hope that you have as much fun and joy with it as I do. Good luck!
