In book VI of the Iliad, Hector says goodbye to Andromache and his infant son, Astyanax, as he heads off into battle. Andromache is an exemplary woman in her society, being a dutiful and caring wife to Hector. For example, she weaves him a cloak and prepares a bath for him as he has asked her as she awaits his return from battle (tragically, he will be killed). She is also a loving mother to Astyanax, which Hector acknowledges as he takes the very young child from the servant and hands him to Andromache.
But "precious" means more than simply that Andromache is a dutiful wife: it means that Hector loves her. This heightens the agony at his having to go into battle and risk death. Although he feels he has no choice—as he says to her, being a warrior is all he knows—the story here reveals the price of warfare. Homer shows us Hector, Andromache, and Astyanax as a family. In a tender moment, when Astyanax cries in fear over seeing his father in a metal helmet with a waving plume, Hector and Andromache laugh, but then Hector takes off the helmet and shows that he is more than a heroic fighter: he is a loving husband and father.
Homer thus critiques the warrior society that he also depicts as heroic: the cost of war is real and heavy, tearing a husband from his precious wife and leaving his beloved alone in the world after his death (her young son is also killed), reduced to a spoil of war.
Andromache is referred to as Hector's "precious wife" because of the position she holds in his heart. Hector is singular in his missions as both husband and warrior. He understands his responsibility to Troy. Yet, he is equally cognizant of his loyalty to his wife. He is devoted to her and their son. His words before he goes on the battlefield is representative of this. Andromache is a devoted wife, one who recognizes what her husband must do, even if it comes at a significant cost to herself and her son. She stands by her husband, even at great pain to herself. She demonstrates her "precious" quality in such unwavering commitment.
Hector reveres his wife. It is for this reason that she is "precious" to him. He never strays in his devotion to her. In a setting where so many men are flippant with their commitments, thinking of themselves over others, and where the Gods, themselves, reveal theseves to be divergent in their wants, Hector stands alone. He is driven by his love for Troy and the love he has for his wife. Both are "precious." Both motivate him. Both stand in opposition to one another and place him in a brutal position of having to choose between equally "precious" but ultimately incompatible courses of action.