Could you give more details? That might be helpful in providing some ideas.

My memory of childhood D&D stuff is very foggy which probably hinders more than it helps.

I think I need to know what the simplest set of character attributes and enemy values might be that can withstand some basic sketch-map-as-you-go exploration and fighting. Is it strength, dexterity and wisdom, out of 20? Or two or five attributes?

Forgot a certain ability on your streamlined ability sheet? Well, who says they all have to be determined beforehand like in DND? Just make them roll new ability specs just when you need them.

Or simply make due with what you've got, e.g. by adjusting what you originally had in mind for the enemy attack with what is possible within your simplified system. And for the next game, slightly update the system whenever you've identified an oversight.

Don't sweat it, the exact rules are not so important, as long as your players can rely on consistency (as a proxy for fairness). For kids, I think, in the end they're gonna love it when they can just roll some dice every now and then and observe how what their characters do results in success when they roll well and (non-catastrophic) failure when they don't. Other than that, I would imagine the theater-of-mind aspect, i.e., the plot and the general world setting will probably be much more important.

We also have a house full of toys so I'm trying to avoid buying more gear. We already have paper, pencils and dice.