Reminds me of something fun we used to do when we had ample free time as kids: We would put random color patterns with crayons on construction paper and then overlay the entire page with black crayon and then scrape through the black to 'draw' with colored fills (the fills being the previously hidden background we made first).
Fingernails work, it's what I used when I did this a few times. Your nails just get really gunky :\ Otherwise, a smooth-edged knife works pretty well. Anything with a decent edge, not sharp. Rounded edges don't work.
Overlaying with black is, yes, covering the whole page with black crayon.
So, the steps:
Fill ENTIRE page with colored crayon patches / squiggles.
Color over ENTIRE page with black crayon. Try to make it TOTALLY black.
Scrape off with fingernail / knife / wooden clay tools / whatever.
Hang on the fridge to admire for several generations.
Ultimately, it's a cheaper way of doing those scratch-drawing things, and doesn't require sharp tools and a careful hand to not gouge the paper.
This is a good technique not just for kids, but for making art in general. I recommend trying it with oil pastels.
Also, the bottom layer need not be colored randomly, but could use a single color or maybe a limited-color palette. Likewise, the top color need not be black, but should cover the colors underneath. Lots of room for experimentation here.
My only recommended change would be that step #4 doesn't use nearly enough black. You don't just want to lightly color over the background, you want to literally put a layer of black wax on top of the background. When you're done, no color should really be showing through.
Try it - it's awesome for kids.