Well, from what I understood from the Trailer, Burnham will be a character with trouble adapting to Starfleet morals... Right ?
Maybe that's why Mudd will have frequent appearances, IDK... He's a Smuggler. He can offer and do things off the books, stuff that the Federation can't. I don't think we're going to see him, as a character, in too much depth. He will serve as the "Space Scoundrel" archetype, a character whose association with Burnham will make those more aligned with Starfleet morals confront her and make her character progress through out the Season.
What I can't guess is the situation in which Burnham (or Captain Lorca, even) will need to do something "off the books", something that requires Harry Mudd's specialty ...
I can be terribly wrong, of course. All we have of Discovery is a 2 minute Trailer.