Andromache is precious to Hector for the following three reasons. First, he knows she loves and cares about him. As she tells him, he is her everything: a husband, a beloved companion, and also, because her father is dead, like a father to her. In Book six, Hector is moved by her appeal to him to stay away from the fighting so he won't be killed, though he tells he cannot do that. When she hugs him and smiles through her tears, he tries to comfort and reassure her as he faces battle. This is a tender moment of marital affection:

she [Andromache] took him [Hector] to her fragrant bosom, smiling through her tears; and her husband was touched with pity at sight of her, And he stroked her with his hand, and spake to her, saying: "Dear wife, in no wise, I pray thee, grieve overmuch at heart; no man beyond my fate shall send me forth to Hades."

A wife like Andromache is also precious to her husband because she is obedient. She does for Hector what he asks her to do, such as preparing a bath for him in Book 22.

Finally, being a good mother was of prime importance in that culture, and Andromache is loving to their son, Astyanax. In Book 6, the small child is with the couple on the ramparts. Hector hands the child from the servant to Andromache, showing his trust for her mothering, and both react tenderly when Astyanax cries in fear because he doesn't recognize Hector in his plumed helmet.

• The first thing would be the husband's feelings toward her.

• The second thing would be the value he places in her as his wife.

• The third thing would be the value he places in her abilities as his adviser and companion.

• The fourth thing would be the confidence he has in her abilities to govern his home and rear his children.

The idea that Hector sees his wife as precious seems to surprise most readers, but women were thought of as very nearly equal to their male peers in Greek society.

Andromache is Hector's confidant and most trusted adviser. He literally can "be real" in the presence of his wife while he must put on the "prince" face when he is out in public or the "hero" face when he is out on the battle-field.  In front of Andromache, he simply is a man.  There is nothing more precious than the ability to be yourself in front of someone whom you trust with your life.