So, first of all, I respect the Arishok because he has the power to take over Kirkwall (according to the Viscount, and later demonstrated by the end of act II), but doesn't. It would probably have been easier to take over/raze the city and accomplish his objective that way, yet he chooses to restrict his army and find the tome (e.g. Ashaad mapping the coast) while trying to maintain a semblance of peace.
On a more personal level, I think it's interesting how he handles adversity. If you tell him that you killed Arvaarad, he's impressed by Hawke's prowess instead of being vengeful, simply warning you not to do it again. If you oppose him by siding with Ser Varnell, he tells you he's looking forward to your death, "as should you." No going after family or blackmailing, á la Meredith.
The Arishok also seems to be the only person who realizes that Kirkwall is indeed a mess, and it's nice of him to acknowledge this -- at times, it feels like no one except Hawke/player sees it. And I know this is a much debated topic, but I also think it's pretty amazing that he's the only one doing something about it. Point in fact, he genuinely believes he's going to improve Kirkwall, as opposed to the nobles and the Viscount, or even Meredith. If nothing else, I respect that intention, even if I'm going to have to disagree about the invasion.
The situation with the elves -- I respect him for protecting them, when obviously no one else could be bothered. If Merrill's conversations are of any indication, as well as the various quests in the Alienage, then there is very little justice to be had for elves in Kirkwall. I think it's telling that it's almost exclusively elves turning to the Qun, and in spite of the fact that the Qunari are not actively converting.
Honesty -- I don't expect him to be up front about matters related to warfare. I actually include the tome in this, because I'm pretty sure someone, be it Tevinter mages or Chantry extremists, would have been all over it, had they realized what it was. However, he gives Hawke a warning when the saar-qamek is stolen, as well as withdraws as promised if he is given both the tome and Isabella.
Furthermore, he has a sense of humor -- he even tells a joke. Twice. In four years time.