Picture a leather box with a lock on it, holding papers that matter. A dispatch is an important official message, so this is the box such messages travel in — you can carry it, and nobody can open it on the way. Everything the word does starts from that box: first the thing itself, and then the famous one that stopped travelling and stayed on a table in Parliament.
From the word’s history
Wiktionary gives the word as dispatch ('important official message sent by a diplomat, government official or military officer') plus box, and records that the Parliament sense comes from ministers formerly using their document boxes as lecterns; purpose-made wooden boxes later replaced them.
Wiktionary 記載這個字是 dispatch（外交官、政府官員或軍官送出的重要公文）加上 box。國會那個意思的由來是：以前部長把自己的公文箱帶進議場，發言時直接當講台用；後來改成專門做的木盒，名字卻留了下來。
Wiktionary, "dispatch box"