How long have you been on the Synthroid? It takes 4-6 weeks for it to reach full potential and it's not unusual for symptoms to get worse before they get better as the body readjusts to having hormones it's been doing without.
Some people do take their Synthroid at night; however, if you're going to do that, you have to make sure you eat dinner early and don't eat any snacks afterward in order to make sure your stomach is empty by bedtime as Synthroid must be taken on an empty stomach. It typically takes at least 4 hrs, or more, for the stomach to empty, depending on what you eat.
It's unlikely to be the Synthroid causing your lack of sleep. You might try taking a calcium and magnesium supplement before bed. They both tend to have calming effect.
T4 which is what synthroid is takes 6 WEEKS to stabilize in your blood. How long have you been on the synthroid?
What are any of your other symptoms?
What are your blood levels for TSSH, Free T4 and Free T3?
This info will help out a bunch?
Some say that you can get a bit of a blood rise in T4 after taking even T4 in about 2 hours. I'mnot to sure about that. But if that is true, then taking the T4 at night would make sleeplessnes worse not better.
It is not uncommon for Hypo patients to have disruptive sleep and to be tired all the time.