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.