The basic thyroid function tests are free T3, free T4 and TSH. Make sure they order the "free's" and not total T3 and total T, which are considered obsolete tests and pretty much a waste of money..
Thyroid autoimmune diseases tend to "run in families", although I believe an actual "hereditary" link is yet to be established. Since you have two close relatives with thyroid issues, I'd also ask for antibody testing to see if you might be developing an autoimmune disease. These are TPOab, TGab and TRab.
These tests will get you started and will let you know if you need further testing, and if so, which direction it should go in.
Best of luck!
Thanks for the insight and advice. I have my appointment tomorrow so I will ask for these tests.