I like a double walled ice bucket because it helps keep the ice from melting, and also keeps condensation from forming on the outside of the bucket.
I never think about ice buckets that I don't think about my grandpa. He misplaced his dentures and no one could find them. They finally located the teeth in the (empty) ice bucket! My grandfather couldn't recall why he put them in there, but he said he supposed it was because he wanted to keep them cool. That's a funny family story that has come down through the years and always makes us smile.