Ok, here...
If you take damage from anything, turret or minion, and you are killed by the turret or minion...
The last champ to damage you at any point in the match gets the kill and gold.
If no opposing champ has damaged you, then the last champ you damaged at any point in the game will get the kill and the gold.
Basically, if you die, either the last champ to damage you or the last champ you damaged, whichever is more recent will get the kill and gold.
You Die, they get credit for it.
I think that is obsessive and complex but I believe it has a spot of possibility.
I may have overlooked something there but you get the idea. No wasted kills or gold.
Hope that makes you happy to play ARAM now.