Here is a brief writeup of the simplified rules I used when I played D&D with my daughters from the time my eldest was age 5 until she was age 15.
Each character basically has three stats: Strength, Dexterity, and Wits. These were largely modeled after the three types of saving throws in 3.5 D&D. At character creation, you assign a d4, a d6, and a d8 to these. They are used for the following things:
Here are some example character sheets. (Click to enlarge.) Note that I drew pictures next to the stats & spells because at the time, they were pre-reading age and needed a visual cue to help them find the right stat when I asked for a roll of some sort.
There are also three derived stats:
Whenever a character had to make a check of some sort (can I climb the cliff, can I avoid slipping, etc.) They rolled the die that was assigned to one of their three primary stats (d4, d6, d8, etc.). Target numbers were typically low (2,3,4), but increased as they gained levels.
If a saving throw was ever needed to resist the effects of an area-effect spell (like fireball, tangling vines, etc.) the target number was half the spellcaster's Wits die.
If they get a mundane item (like a lantern) it does what they would expect.
If they got a weapon, I needed to indicate what kind of damage it did. Basically, small weapons (e.g. dagger, small bow)=d4 damage, medium weapons (e.g. sword, medium bow / crossbow)=d6 damage, and large weapons (e.g. battleaxe, spear)=d8 damage.
If they get a magic item, that's basically the equivalent of getting a new spell. (I wrote a similar table on the reverse side of their character sheet for magic items so they could see what their options were.)
A hallmark of my play experience with my daughters was finding four-leaf clovers. These would basically grant rerolls when they needed them (one roll per leaf). It was amazing just how often they seemed to be able to find four-leaf clovers during their adventuring.
They often found gold pieces and gems that they could use to buy equipment at a shop.
Many years after I invented this system, I discovered USR ( "Unbelivably Simple Roleplaying") and found that I had basically re-invented this same wheel. Savage Worlds uses similar dice mechanics to the ones I've outlined here.