A brainbox is someone very clever — the classmate who knows the answer, the friend who undoes the puzzle, the person whose head seems full of smart thoughts. It is informal, and it sounds especially British; a teacher in London might say it with a smile. It can praise (“our little brainbox”) or tease a little, but it is usually warmer than nerd, and much kinder than brainless. You would not put it in a very formal report. Americans more often say genius, brain, or brainiac. Though actually, it does not mean the person has a different kind of brain; it is just a friendly way to say they think quickly and know a lot. Older, and less common now, brainbox can also mean the head itself, the “box” the brain sits in — “use your brainbox” the way we say “use your head.” It is not a real box.