Barb's advice above is good. Optimal thyroid levels are also critical for his physical and mental development.
Test Ft3 and FT4 and adjust medications until his numbers are mid-range. If FT3 is still low even with adequate T4 levels, then he may also need to be treated with T3 medications.
Unfortunately, TT4 is considered obsolete and of little value. His doctor should be testing Free T4 and should also be adding Free T3, which is the hormone that's actually used by the individual cells, which Free T4 is a storage hormone that must be converted to Free T3 prior to use by the body. Without those 2 tests, there's really no way of telling exactly what his thyroid status really is.
Of the TT4, approximately 95% is bound by protein and is unusable, which is why the Free (unbound) portion needs to be tested, as that shows what's available for conversion to T3. As with TT4, most of the T3 is also bound by protein and can't be used, so again, we have to test the Free (unbound) portion to determine what's available for individual cells to use.
Being unable to sit still or not being overweight are not good indicators of having the opposite of hypothyroidism, particularly with increased TSH. Many symptoms can apply to either hyper or hypo and not everyone with hypothyroidism gains weight. There are many other symptoms of hypothyroidism that we often don't even connect with it, including, but not limited to, fatigue, constipation, depression, brain fog, muscle/joint aches/pains, thin, brittle hair/nails, thinning eyebrows, sensitivity to cold, etc. We might not, readily, notice some symptoms in someone who can't communicate well, with us. One of the first symptoms to go away, for me, was puffiness around my eyes, and I hadn't known that was thyroid related.
What's your objection to having your son take thyroid hormone? Behavior would only be one indication as to whether or not he should be placed back on the synthroid. Has he been tested for thyroid antibodies to determine if he has Hashimoto's or has there been another cause of hypothyroidism determined?
While TSH is only an indicator and should not be used as a sole diagnostic, the elevated level is an indicator of a problem and should certainly be taken seriously. Anyone not having adequate thyroid hormones for an extended period of time risks very serious health issues, even if they aren't displaying obvious symptoms or seem to be functioning normally.
Prior to being diagnosed with hypothyroidism and later Hashimoto's, I didn't complain and I never missed a day's work, though I felt like death warmed over for years prior; if you'd asked most people, they'd have told you that I didn't display any symptoms of an illness. The fact was, I had them all; I simply didn't discuss them and I managed to power through the whole ordeal; however, had it gone on much longer without a diagnosis and medication, I would most likely have found myself in the hospital with myxedema, which is severe untreated hypothyroidism and my road to recovery would have been much longer than it was.
We always recommend treating hypothyroidism sooner, rather than later because once symptoms get away from us, it's much harder to get them under control.