Honor is important to Hamlet to the extent that even though he doesn't want to have to commit murder, he is willing to do so because his father's ghost asked him to do it: "Oh cursed spite, / That ever I was born to set it right!" (the end of Act 1, sc. 5). This is what Hamlet wrestles with throughout the play: he has promised his father's ghost he'd seek revenge against Claudius and he's frustrated with himself that it takes him so long to carry it out. Honor drives him to make the promise and honor drives him to ultimately make good on the promise. In Act 2, sc. 2, in his "rogue and peasant slave" soliloquy he laments his inaction. In his "To be or not to be" speech in Act 3, sc. 1, he again laments his inactivity blaming it on a fear of damnation. In Act 4, sc. 4, the soliloquy that ends that scene expresses Hamlet's determination to stop dragging his feet and do the honorable thing, which is of course to kill Claudius. Laertes also considers honor important. In Act 4, sc. 7, Laertes wants vengeance for his father's death at the hand of Hamlet. He is willing to do a dishonorable deed - cheat in a fencing match - in order to get that revenge. It is difficult to say which character had more honor. Laertes cheats, but Hamlet has Rosencrantz and Guildenstern put to death simply because they allowed themselves to be pawns to Claudius.
Hamlet is heir to the throne of Denmark. That position, in and of itself, demands that Hamlet understand and seek honor. He is none too pleased with the situation of his mother and uncle wedding one another in the wake of his father's death, as the marriage is less than honorable, at best.
Laertes, on the other hand, is more concerned with the honor of his sister remaining intact, warning her, in essence, that Hamlet might not be a stable fit as suitor. He says all this, of course, out of concern for the family's reputation and, again, honor.
After all, Hamlet has been wallowing in his black-bile melancholy because of his father's recent death, and due to the marriage of his uncle Claudius to his mother, as previously mentioned.
As Laertes heads back off to college, Hamlet remains and the plot unfolds, showing us how the young royalty truly seeks honor in the midst of a great Shakespearean soap opera.