What does the word "honor" mean here?

It means that they don't acknowledge the good qualities which are attributed to Cassandra. It means they don't "pay tribute" to her talents. If you were a gifted musician, i might honor your talent simply by telling all of our friends that you are a gifted musician. In this case, it simply means to take notice of, to acknowledge.

Respect.... Honor demands respect... To show respect... Deference... Honor is respect... Honor is self respect... There was once an honor economy just as now there is a money economy, and there were honor societies just as there is now money societies... It is a vestigial word whose meaning has become obscure mostly because where honor is dear, money is rare; and where money is dear, honor is cheap... And because so many claim honor who have none of it, that it has become all form and no relationship, and no relationship means no meaning...
The book of wrath, the Illiad was an affair of honor... It was not Helens face that launched a thousand ships, but honor; and all those there were there in fulfillment of a mutual pledge of honor... Agamemnon is your typical blunt object, and his honor is often called into question; but when Achilles honor is insulted, he goes to the ships and weeps, because honor held him there even while abused, so honor held him out of the fray... But when his friend is killed, honor once more brought him back into the fight, and when warned that the death of his victim would be followed by his own he replied: Prepare me for my death....

The original Cassandra warned Agamemnon, and brute that he was, he did not give credit to the Gods for his victory, but marched unwashed into the temple, inviting destruction... His wife acted for honor, and so did her son in killing her.... Our word rehabilitation gives some hint of the meaning, since none could live in primitive societies without honor, and home, habilus was honor... So people demanded that the criminal be driven out, or killed for fear of the reprisals of the gods, and only a family member, to have the family honor could do it... We simply cannot grasp from our vantage point in time what honor means, and once meant.... Now it is just a word, but to them it was as dear as life, and as essential as life, and now people ask: What does it mean...

To truey know what honor means you must endure for it, and work for it, and buy it at a high price.... Short of this, honor will only mean acknowledgement, or resepect... That is but a fraction of its true meaning...