Well, WoW had a nice solution for it. Whenever an item dropped for you in a dungeon, but you couldn't pick it up (due to a full inventory), the item would be sent to your mailbox instead.

I can fully understand the frustration of having stuff hit the floor because of a full inventory. I had that the other day when I was doing the Kobold Beastmen quests. I needed to get the Alchemist Fate drop for one of my quests, but even though I made room for the fate reward, it still hit the floor, due to monster loot filling up the spare slots. Luckily the fate respawns quickly, unlike the Kafra (the Boar in East Shroud).