Autoimmune Thyroid disorders are often connected to Celiac. See theglutenfile.com
for more info.
Those with Lyme should ideally avoid pro-inflammatory proteins found in gluten and dairy. Nightshades may also need to be avoided.
Consider also the possibility of parasites (the most common co-infection with Lyme, but can occur in the absence of Lyme as well), and Candida overgrowth.