My ideas to improve Honor
Once you get an honor ribbon, no matter which one, you can enable/disable your summoner icon appearing on your champion. I'm not sure exactly how this would work or how hard it would be, but just imagine your summoner avatar on Ashe's cape. You know the little circle holster that holds Shen's blades? Imagine your summoner icon on the middle of that circle. Talon would have your summoner icon on each of his little mini-capes. Lucian could have it on his pistols.
After getting multiple ribbons, you can choose which one you want to display.
Once you get 2 ribbons, your summoner icon not only shows up on your champion, but also the ribbon you currently have chosen to display. The ribbon would be on Ashe's bow, or on Shen's arm like an armband.
Once you get 3 ribbons, you get some free RP and an IP boost. Additionally, the color of your text in chat (whether it's to friends, in your status, in pre game/post game lobby, chat rooms, in game) is the color of the ribbon you currently have displayed. Honorable opponent means your text is red, etc.
To help avoid the general abuse of honor, you get bonus honor points if someone you have recently honored gets a ribbon. (Say, 1 week) if they don't get a ribbon, you lose out on that bonus honor. If you honor everyone, you lose out on all that honor you are giving. The best way to make sure your honor is spent is to give it to truly positive players, so that they get ribbons, and you get bonus honor. If you honor someone who already has a ribbon, you get a very small amount of bonus honor.
Mr Bl3u says we could also have borders similar to the elo borders in the loading screen that also appear in game on the tab scoreboard. I'll iterate further on this: To prevent honor borders from clashing with rank borders, honor borders won't actually be borders - they will change the whole card. If one is an honorable opponent, their card in the loading screen will be red. If they're gold elo, the red card will have a golden border.
Solari Eclipse says we could have Team Fortress 2 style customizable accessories for getting honor. I'll add to this: Instead of just hats, each champion would have their own accessories. Draven could get mustaches on his axes if the summoner has a certain amount of honor, or Shen could have katanas instead of just swords. These changes would be minor enough to be added to skins, so it's not like only Classic Draven could have mustaches on his axes. If we want to get really crazy, we could even color the accessories according to the skins. Soul Reaver Draven's axes' mustaches would glow blue instead of brown.