The average of the average number of honors per game is slightly below 1. So there aren't really any sure fire methods of getting more honors. Aside from that, receiving honors is one of the smaller ways that honor increases.
The primary way to get honor is by playing games without being validly reported.
This means that avoiding everything that can be flagged as negative is a good idea. No "pointing out" anyone's KDA or CS. Don't say that you're going to report or mute anyone. When you decide that someone is trolling/griefing, stop interacting with them. Don't blame people for the state of the game.
The second most important way to gain honor is by using chat positively. This can start with simple things like "gl hf" at the beginning and "gg" at the end. More advanced techniques include using chat to help your team win the game. Make plans for the future. "We should get a ward on baron" "Take dragon while they're distracted" "Build XX next for some MR" "focus LeBlanc first next fight" "Hold your ult until after they use their dashes"
While you are doing this, make sure that you're talking about the future. The future can be changed, but the past cannot. "Why didn't you ward baron?" "You should have taken dragon while I had them distracted" "Why did you build XX?" "stop focusing the tank" "stop wasting your ult" Those, while essentially the same message as the prior ones, are flagged as negative because the past can't be changed.
The third most important way to gain honor is the team bonus for everyone honoring someone. This means that it's best for you to honor someone every game. This also helps you to enjoy the game itself more. It focuses our thinking on what went right, and who we enjoyed playing beside the most, instead of focusing on the troll, flamer, or uncooperative person.
The smallest part of gaining honor comes from the individual honors. Focusing on those is a good way to make yourself frustrated.
