Homer uses epithets to tell us something about his characters, be it personal qualities—such as "fleet-footed Achilles"—or their functions within society—e.g. "Hector, breaker of horses." He also uses these epithets to pad out the line, to make the rhythm of the verse flow more smoothly when spoken out loud. The Iliad, like Homer's other masterpiece, The Odyssey, was originally intended to be sung, and the poet's copious use of epithets greatly enhances their musical qualities.
In the case of Andromache, the epithet "precious" is an example of foreshadowing. Hector knows what will happen if the Trojans should lose their bitter war against the Achaeans. All the men will be killed, and their wives and children sold into slavery. That being the case, one could argue that Andromache is precious in much the same way as an antique vase is precious: vulnerable to being broken. That would certainly seem to be the best interpretation of Hector's understanding of the word, as can be seen in the terrifying vision he recounts in Book VI of The Iliad:
"Thy griefs I dread: I see thee trembling, weeping, captive led!"
Hector's seen the future and it chills him to the bone. He sees his beloved, nay precious, Andromache, being led away in chains, a sex slave of the victorious Achaeans.