Item Management
Currently the items not equipped to heroes are stored on the left, just below your character. Frustratingly, the partial set buffs dominate this space, obscuring the items and popping up their own tooltips when you try to hover over an item to see what it is. Since there's not much time to prepare, and you may need to do many other things in that limited time, this UI bug has significant negative impact on gameplay. To solve this, I propose to move unequipped items to the strip of grass between the benched champions and the shop.
Also, it's currently only possible to unequip items from champions by selling the champion, which creates a perverse incentive to put items on champions you don't want to keep, until you eventually luck into getting a champion that you want to have that item as part of your long-term strategy, even if in the short term, you'd rather the item be used by a champion you want to keep, but not be stuck with that particular item for the whole game. To solve this, I propose that you could right-click a champion to open the tooltip, then either left-click the item to pick it up or right-click the item to return it to storage.
A related issue is that if you get 2 items you want on the same champion, but you want to combine them with other items, you can't put them both on that champion until you find the item you want to combine with the first item given to that champion. To solve this, I propose that instead of automatically combining items given to champions, they fill up empty inventory slots by default, but you can right-click the champion and open its tooltip, then place the item directly over the item you want to combine. However, because this proposed UI would not make it obvious that items even can be combined, some additional feature is required to communicate that, e.g. some item combination tutorial. However, the best way to communicate this would be...
... to provide an in-game pop-up grid showing the various item combinations and their results. Currently players with multiple monitors have an advantage in that they can have the LoL client running on their primary monitor, while browsing a list of item combinations on their secondary, while players with only a primary monitor have to either wait several seconds while switching between fullscreen LoL and Windows with their browser. This is important information players need to know before the next draft so they can identify which items they want to rush to claim.