Cortisol seems to be a big player for those taking T3/T4 thyroid meds (or no thyroid meds at all). It has been my experience that those with lowish 8 am cortisol values (in the lower 50% of the range) tend to suffer hypothyroid symptoms. Even if T3/T4 look decent, TSH is sometimes still screaming for more. This is usually accompanied by higher Reverse T3 (RT3) issues, which is basically pooling of T4 hormones that are not able to convert to T3 (which are the metabolically active forms).
So if your friend is not seeing expected results despite being on thyroid meds, cortisol testing is probably warranted. I've seen people correct high RT3 issues with a combination of lower dose Cortef hydrocortisone (20 mg/day in divided doses) and T3 only meds (dessicated thyroid). Some may be able to benefit from switching to dessicated thyroid alone with the added cortisol support. Blood testing would give the proper treatment path.
Note: I do not advocate doing this method alone, though it oculd technically be done using research chems and Isocort in place of hydrocortisone. I would never do it without my doctor's knowledge.
Other hormones and bloodwork that seem to be strongly correlated with thyroid are Vitamin D, iron, ferritin, and iodine.