Is there any point to honor the friends I play with at the end of every League game? It seems like the system would be totally broken if those 'points' counted just the same as those from multiple, random partners. Does the system just discount their worth heavily, or into oblivion?
Actually it's not pointless and honoring a friend isn't against the rules. "Honor trading" was rampant when the honor system was first added to the game, but not because of friend's honoring each other.
It was rampant because players would trade honor by telling teammates and opponents alike that they would honor them in return for being honored. Regardless of them being good teammates/opponents, being fun to play with, or the player being helpful by offering their knowledge in a constructive way to teammates.
This is what led to many players having the honor badges very early on when they would not have otherwise (trolls/bad players) and many of them were punished by RIOT for this via bans and the loss of the honor they had accumulated up to that point(this included the badges as well).
In regards to the effective weighting of honor from friends vs. players you don't know, it is calculated with a few things.
1) How often the player that is honoring you, honors players in general?
Say that all four of your teammates honor you for the same thing, "Teamwork". Now three of those players consistently honor other players every game, they enjoy playing with people and what have you. But, the fourth guy on your team rarely honors people, and in fact you are the first player they have honored in over a week. The honor you received from that fourth teammate is effectively worth more towards your honor badge then the other 3 combined. This is because the more you honor the less your honor is worth.
2) Is the player that is honoring you a friend or a player you were matched with randomly?
This one is fairly straightforward. As most of us know, playing a game with random players can be a crap shoot, and as such, having a random player enjoy the game they played with you and rewarding you with honor will generally have a heavier weight towards your honor badges than your friends that you play with everyday. It is still possible to get honor from a friend that is worth more than a random player's honor, but this would take into account factors from the first part of the calculation as well.
So in conclusion, honoring your friends is perfectly okay. It is NOT worth it to do it every game you play with them due to the reasons given above, but honoring a friend you haven't played with in a while can still positively increase their honor towards the goal of getting an honor badge(again, after the factors have been calculated).
Even with 58 Teamwork Honor, I have still yet to receive the Teamwork Badge(even though my brother received his with far less honor).