The story
From dispatch (“important official message sent by a diplomat, government official, military officer, etc.”) + box. Sense 2 (“box placed on a table in a legislative debating chamber and used as a lectern”) is from the fact that in the Parliament of the United Kingdom dispatch boxes used by ministers and other Members of Parliament to carry important documents (sense 1) were formerly used as lecterns; these have generally been replaced by cases, usually made of wood, specifically made to be used as lecterns.
Etymology from Wiktionary, CC BY-SA 4.0.