Honoring the dead
Both sides in the Trojan war hold the same basic religious world view. They honor the same gods, for instance, and they both understand that funeral rites are an important part of the dead person's ability to succeed in the afterlife, so when Priam is unable to recover Hector's body and give him a proper burial, it is a problem of epic proportions—literally. The main symbolic representation of this theme is the gods' prevention of Achilles from defacing Hector's body. This is their way of honoring Hector for his valiant warfare.
Simplicity as useful and good
There is a moving scene in the novel where this theme shines through very clearly, when the simple man Somax is chatting Priam's ear off about his family and his daily life. Priam is charmed by the meaning and depth contained in the man's simple life. When Priam himself, towards the end of the narrative, chooses simplicity and humility, the gods themselves bless his quest.
Reconciliation as mutually beneficial
The novel is careful to depict the ransom exchange between Priam and Achilles as mutually beneficial. Achilles didn't know it, but he was also in desperate need of the exchange, because his repeated failure to deface Hector's body just reminded him of his dear friend Patroclus. In order for him to mourn Patroclus properly, he has to understand that the family on the other side of the war feels the same emotions. In other words, the communion with the enemy grants him freedom from his agony. In that way, both parties get something important. And of course, Achilles is also enriched materially by the king's gifts.
Aging
Throughout the novel, we are made very aware of the age of both Priam and Achilles. Achilles being young and in his prime is juxtaposed against Priam's being old and in the later years of his life. But instead of age being a bad thing, aging is seen as a privilege, and being old is a mark of wisdom. The tragedy surrounding Achilles is partially his knowledge that unlike Priam and Somax, he will never get to grow old as he should have. For Priam, the authority that comes with age is a large part of what enables him to conduct the ransom exchange and get back Hector's body.
Relationship between father and son
The relationships that form between fathers and sons are at the heart of this novel. The book is about a father ransoming his son's dead body. Priam struggles with the grief he feels for his son, and connects to Somax as a result of shared grief that is specific to fathers that have lost sons. Achilles feels a longing for both his father and his son, and is moved to help Priam because of his knowledge of how torn he would feel in Priam's place, as well as his understanding that he would not want his own father to be in Priam's place.
Similarities across differences
Although the novel is about a war, the characters often find points of contact across sides. Achilles and Priam find similarities in being fathers, as do Priam and Somax. Priam learns to identify with the simple carter, and Achilles is forced to consider the perspective of the father of his greatest enemy. All of them are united in the fact that they have lost someone they love and are not sure how to cope in the aftermath. Despite not being on the same side or from the same circumstances, they are able to understand one another, even if only for a moment.
Dying
The threat hanging over both Achilles and Priam is the threat of death. While the central conflict is about honoring the dead, both Achilles and Priam are very aware that their time is coming. For Achilles, death comes to him as a young man and on the battlefield, a prophecy that Hector shares with him as he is dying. Achilles' arc in the novel is defined by his knowledge of his fate. Priam deals with dying because of the fact that he is an old man who knows that he is at the end of his life. His arc is defined by his desire to do something great before he dies, as well as the desire to bury his son Hector properly.
