I'm in the process of working on this myself, specifically for my soon-to-be-seven year old, his six and nine year old cousins, my brother and our father.
I'm probably going with Labyrinth Lord or Beyond the Wall. Both are simplified rulesets (LL emulates the BD&D of the early 1980s, including non-human races effectively being their own classes, while Beyond the Wall is meant to emulate young heroic fantasy novels like the Black Cauldron or a Wizard of Earthsea).
Since my boy views Minecraft as the lens through which all games (and all of real life) is viewed, I'm picking up some rhinestone crystal hearts this month to serve as markers for hit points (my brother first played AD&D at age 8 and freaked out when he died -- I want the boy to have a visual reminder of how healthy his character is) and another sort, maybe diamonds, to be used to show magical resources.
I'm also picking up some plastic Reaper Bones minis and getting the old Dungeon Tiles while they're still available, as playing the Dungeon! board game with him has shown me how valuable to him seeing the map and monster miniatures has been to help him visualize what's going on.
For kid players, I recommend printing out, and possibly laminating, pre-made characters ahead of time, maybe just leaving a spot for a name and a picture blank for the kid to customize the first time out. (My boy's older and ahead of where your kid is.)
I'd also come up with some adventures that don't necessarily involve combat: Finding lost animals, children who've been "kidnapped" when they went off to play with fairies or discovering where a lost treasure is are all a lot less frightening than many D&D adventures are, while still being fun.
I'd also figure out what monsters and perils are too much for the kiddo -- my kid isn't scared of zombies, thanks to G-rated computer games featuring them, but monsters that seem like scary people (orcs, etc.) are too much for him. I'm likely going to have cartoonish frog men be the foe in their adventure while on vacation this summer, for instance.