I think it’s common in pregnancy for women to feel this pain. It’s actually a form of carpal tunnel. It happens due to the swelling and goes away after you deliver. I had it too.

It may be related to pregnancy. Pregnancy can cause your tendons, muscles, etc. to relax. If you use your hands frequently, such as for typing, etc., the loosened ligaments can allow you to cause more damage and more pain. Hence why carpal tunnel is indeed common in pregnancy. [It's also recommended, for this reason, that you're far more careful while working out, never trying to actually build muscle, but instead maintaining what you have.]
It wouldn’t hurt to ask your doctor what s/he thinks, or get a friend/significant other to give you a hand massage.