Luckily, there are no missing kidneys involved in this story. But the ice bucket is a great tool for, well, keeping things cooled overnight if there's no fridge available.
Personally, I've got a mind for some yogurt in the morning, so often I'll fill up the bucket with ice and close the container before going to bed. In the mornings, the contents in the container are still cold, and there's still creamy dairy goodness without worrying about it spoiling. This also works well for fruit, small individual cartons of milk and other small refrigerated items.
If you have more items, sometimes it's possible to request a refrigerator from the front desk for a nominal fee. If your room has a minibar, I would be careful if you choose to use this as a way to store your items. Apparently, some minibars contain sensors, allowing hotels to figure out if some items have been shifted around. There's no way you want to be charged $10 for moving around some M&M's!
(Editor's note: We didn't know some minibars have sensors! Wouldn't life be just simpler if everything was free?)