FSH has to do with her ovaries (follicle stimulating hormone). TSH (thyroid stimulating hormone) is the one connected with the thyroid gland. If the TSH and free T4 levels are normal (and different people consider different things normal), then she isn't suffering with low thyroid. The supplement she takes replaces what her own thyroid doesn't make, so her thyroid hormone levels are normal and she should not be having any side effects from that one way or the other. Too much replacement hormone can interfere with sleep.
Melatonin won't hurt her or interfere with other medications. However, if she is waking so often during the night, she isn't getting her proper rest and that is why she is sluggish and tired during the daylight hours. Her physician needs to work on why she isn't sleeping properly. He may want to order a sleep study, or consider other things that can cause poor sleep.
Some things she can do on her own are to avoid anything with caffeine (coffee, tea, cola drinks) after 3 p.m., avoid eating within 2 hours of bedtime. A glass of wine with dinner won't hurt, but too much alcohol can also interfere with sleep. Avoid bright lights and stimulating TV, exercise (other than sex), etc. for an hour before bedtime. Sexual relations can actually help her relax and get a good night's sleep. Exercise during the day will actually help, if it is no more than walking around the block for 30-60 minutes. It's also a great way to wind down after a stressful day at work. If she is using the popular "energy drinks" that should stop. While they give a boost during the day, that can boomerang at night.
I hope she will discuss this with her physician if simple things like changing eating habits and exercise pattern don't help. There are dozens of things that can cause less than restful sleep, but sometimes it takes some sleuthing to tease out exactly what the problem is in any individual case.
I hope this helps!