At the least it would be worth seeing if the removal of wheat and other gluten-containing grains helps. Dr. William Davies, who has just released a book on the (many) problems caused by wheat, has amassed quite a bit of evidence that it does in a very large number of cases. He has thousands of people pass through his practice, and he says that whatever their medical problem, eliminating wheat usually helps.
Wheat Belly | Lose the Wheat, Lose the Weight
It might also be helpful to eliminate all soy products and all products containing refined carbohydrates (sugar, high-fructose corn syrup. etc.). The same might go for dairy products—at least unfermented ones.
That, in a nutshell, pretty much is the Primal approach. Paleo is much the same. I'd suggest you drop an email to Rob Wolf—if anyone has heard directly from parents of children with this problem it's probably he:
Robb Wolf | The Paleo Solution book and podcast | Paleo diet, Paleolithic nutrition, intermittent fasting, and fitness
With people who have more intractable conditions something like a GAPS diet might be necessary. Dr. Natasha-Campbell, who devised the diet (it's based on the SCD with a bit of a WAPF twist) uses it with people with a range of psychological problems that seem to have a link to a lack of gut health (autism, ADHD/ADD, dyslexia, dyspraxia, depression and schizophrenia) and is currently writing a book on the range of physical conditions in which it can also help. Here's her site:
GAPS Gut and Psychology Syndrome (GAPS) - Natural treatment for autism, ADHD/ADD, dyslexia, dyspraxia, depression and schizophrenia