This is a term that seems to be in very in vogue right now. Every time you gather three or more officers in the presence of their reporting senior this term is sure to be bandied about. It calls immediately to mind a hardy constitution, a selfless nature and a servant leader.
Therefore if you proclaim that you are doing 'missionary work' you immediately frame yourself in a very specific way in the mind of your boss. He sees you as someone who 'reaches out', who 'socializes' ideas; ideally he sees you as an "early promote". Beyond that he sees you as someone who works to convert the hearts and minds of other agencies that don't normally work with the Navy or who aren't traditionally fans of the department of defense.
Examples of use:
"Yes, Admiral I've been doing serious missionary work over at A-I-D, and I think it going to start to paying dividends at our next conference."
-LCDR X at any given weekly 'what have you done for me this week' meeting
"LT, I need you to start doing some missionary work at the Dunkin Donuts, I think they've been shorting me on my donut hole orders."
-Flag officer X