I would not wing it; call the other child's mom and ask her up front if it's OK and if so, what you should make. If the kid's allergy is not life-threatening you might be OK (again, ask her first) but If it were me, I would not even try. And if you make a nut-free, eggless treat and carry it over, but it turns out the mom is worried about serving it, you'll end up with two disappointed and frustrated kids who see a snack come in the door but are told they can't have it.
We can give you a thousand recipes for nut-free snacks but -- for instance-- if the child has a severe peanut allergy, and you make a nut-free snack but happen to use a cookie sheet or spatula that you used a while back for PB cookies -- even a remnant of PB oils on that sheet, however clean, could set off the allergy. (It's true, if the allergy's severe.) And do you really want to pore over every label of every ingredient to ensure it was all made in nut-free facilities and contained no egg or egg byproducts?
It's truly thoughtful and nice of you to want to take a snack but I'd either purchase a packaged one that is especially for peanut-free, egg-free consumers, or I'd skip it and take the mom a surprise mocha or whatever from a coffee shop that's on the way to their house....Many moms would love that!
If the point is mostly to keep your kids busy before the play date, find another activity like a craft; they could make a thank you note for the mom in advance, or a picture for the friend.