Is there a measurable benefit to feeding kids organic food?

Yes. I usually tell people to eat "god made" food and leave the "man made" stuff alone. No need to get food from the organic section in the supermarket. Stick to the "ouside wall" where you find the fruit and vegetables. Avoid all the sugary junk, soft drinks and commercial fruit juice. If the food that we were intended to eat had not been here before we arrived, we could not have survived.
Yes. Modern farming methods, heavily reliant on chemical fertilizers and pesticides, are producing food very different than the food our ancestors ate. Plenty of evidence that modern farming methods are unsustainable, have a heavy carbon footprint, and use a lot of fuel. Research problem on diet: so many variables! but no doubt organic foods are good for your kids' (the planet's) future!
Sometimes. It depends on the food. There is a great benefit in feeding children certain organic fruits and vegetables because some are high in pesticide content. Go to www.Ewg.Org to find a list of the "dirty dozen" which are fruits and veggies with the highest pesticide content.
Organic foods/kids. Some foods are better fed as organic. Soem foods are just overpriced as oprtganic e.G mangoes. Who is going to climb up on mango trees to spray the fruits? Mango trees grow well without any fertilizers. Us eur common sense for what is safe if not organic. Dont let the industry scheme u.
Don't Know Yet. Altho it "sounds" better, there is no research proving an organic diet is better. We will see...
Measurable, no. Doesn't mean there is none; just none anyone can quantify.