to be heavy: to weigh much, whether a literal load, a crushing trouble, or a body part that has grown too heavy to work
This is the plain, physical root of the word: something has real weight, or so much of it that it presses down. From there it spreads in two directions a reader can feel. A weight that bears down on a person becomes a burden: a hand laid heavily on someone, a famine or a war that grows severe, sin or labor too heavy to carry. And a part of the body that has grown too heavy to function becomes dull and unresponsive: eyes too heavy to see, ears too heavy to hear.
The hardest case sits here too. When the word is used to make a heart heavy, it does not mean sorrow; it means the heart has become like a dead weight, unresponsive and unmovable, the picture behind the hardening of Pharaoh. The same image of dead, immovable weight runs through all of these.
This concrete sense of weight is also the seed of the word's other meanings. A person who carries weight in the eyes of others is honored, and to treat someone as weighty is to honor them. Those uses are separated out below, but they grow from this same root idea of something that has heft.