Borrowing a bit from my childhood, a bit from various scout manuals, and a bit of what I've done with younger cousins, and friends and their kids.
Call it a "woods" kit, and have them bring it with them whenever you head into the woods (hiking, camping, chasing the dog, whatever). Take yours too, and make a point of calling it your "woods kit". Keep the focus on fun and learning. Spending time to teach them is going to be more important and more useful than just handing them a bag of stuff.
Start with
- A bit of cordage, whatever type you like; it's going to get tangled and knotted and otherwise "used up" but that's OK, that's called learning.
- SAK of some sort is a good choice, if they're responsible enough to carry a knife (I know some 10 year olds I'd trust with their own knife, and some 20 year olds I wouldn't -- age isn't everything, maturity is).
- Small flashlight -- AA LED Mini Maglights are damn tough, readily available in a bunch of colors.
- Water bottle, with fresh water in it.
- Whistle
Let them pick colors, and what/how they want to carry things: waist pack, haversack/bag, backpack. Teach them simple (but useful and fun) things, like how to tie sticks together to make useful bits. Teach basic knots. Show them how to carve sticks -- making a pointy stick is the same process, regardless of the end result being used for a tent peg or a hot-dog / marshmallow cooking stick.
As they get older and their interests change (and hopefully, grow), you can add to their "woods" kit.
- Fire kit: Start with a Bic and teach them how to build a fire lay. Show that it's useful outside the woods too, like starting a charcoal bbq grill. Ferro rods and bow drills can wait until there's an interest. Keep things easy until they want more.
- Shelter: Small tarp, 4x6, 5x7 at most to start with. Keeps them from being frustrated with their tarp always turning into a kite.
- Different types / more / better cordage, as needed.
- Fixed blade knife (again, if/when they're mature enough)
Add more skills -- Use their basic carving skills (pointy sticks) to make feather sticks and shavings for a fire lay. Useful knots and lashings extends to making a tripod for over a fire. If they're up to it, carve notched sticks or find suitable forked sticks lashed together to make a pot hanger. Take the fabric seat off cheap chinese tripod stools, add that to their pack, let them find/cut lash their own seat legs together, and teach them why they should use good cordage and sturdy sticks for things that hold them up.
At some point, you might need to upgrade the container their "woods kit" is in. Maybe they've outgrown it. Maybe they've worn it out and it's falling apart. Maybe they've decided that not everything needs to be unicorns and sparkles. Maybe they've just decided that they hate the color blue now, but green is awesome. Roll with it as best you can.
Eventually, they're more than likely going to lose interest, but at least you've given them the basics in terms of skills and equipment.