For what reason did the Dali Lama's traditional daily schedule have washing before meditation? I often hear that meditation is best at the first part of the day, and it is usually recommended to meditate before eating. Yet it seems odd to me that for the Dali Lama, washing would come first and meditation would come second. 

The only explanation I can come up with is that the washing would allow for enough time to pass so that the early morning 'dreamy' state of mind would wear off?

(I am not sure if the Dali Lama still follows his traditional schedule, I am talking about the Dali Lama of centuries past)
Washing is a symbolic cleansing. One finds it in the Tibetan precepts ceremony. The modern Dalai Lama washes his hands before reading too.