Ah, Scotland. Walking in amazingly beautiful glens, trying (and failing) to decipher what the locals are saying, and scanning the mountain tops for golden eagles (that remained elusive till the end). This last pastime during our holiday in stunning Scotland reminded me of this expression. I find it very expressive: like a hawk watching its prey with its unblinking piercing eyes, I focus all my attention on one thing.
Only this is not what the expression actually means. I read a short piece in the Independent a few weeks back that told us that this saying refers in fact to the practice of old-fashioned falconers who used to "watch a bird" as part of its training. The man would stay awake longer than the bird could, until, after many hours, the bird at last consented to go to sleep on the fist. The hawk is being watched, not doing the watching.