Hector's own words in Book VI demonstrate how precious Andromache is to him:
And yet no dire presage so wounds my mind, My mother's death, the ruin of my kind,  Not Priam's hoary hairs defiled with gore, Not all my brothers gasping on the shore; As thine, Andromache! Thy griefs I dread: I see thee trembling, weeping, captive led!
Here, he expresses his dread fear that something terrible could happen to her in the event of his death, such as being taken prisoner (which is indeed her fate). In his mind, leaving her to her fate disturbs him more than losing his mother, his father, Priam, or his Trojan brothers. Implicit in this quotation is that their marriage was more than strategic; Hector truly loved Andromache.
In return, Andromache expresses her love and devotion to Hector, when in Book VI she expresses:
"Yet while my Hector still survives, I see My father, mother, brethren, all, in thee: Alas! my parents, brothers, kindred, all Once more will perish, if my Hector fall..."
It is clear that she puts her marriage and love for Hector ahead of her love for the rest of her family. To have a wife that is more than a function of a political marriage and the producer of an heir makes her precious to Hector.
Lastly, in this male-dominated cultural context, women were often the spoils of war; one needs only to consider the importance of Helen, the wife of Sparta's king, Menelaus, whose abduction sparked the Trojan War. Women were chattel, and in this way Andromache was precious, not just to her husband, but also to his enemies. After the Trojan war and Hector's death, Andromache was given to Neoptolemus as his concubine, and she gave birth to his son. Women in this culture were objectified, and Andromache was no exception to this truth, but Homer does convince the reader that her value to Hector was more than that of chattel.