The OED (228) defines this as a sailor’s perversion of midshipman. The latter is defined as “in the navy, the designation of a rank intermediate in the line of promotion between that of naval cadet and that of the lowest commissioned officer (i.e., in the British navy that of sub-lieutenant, in the U.S. navy that of ensign).” Another definition is “a diminutive midshipman” (115). As traditionally cast, the midshipmite is indeed a little mite of a lad –– a Gilbertian visual pun.