The notion apparently is of being "puffed up" with pride; compare Old English belgan "to become angry, offend, provoke," belg "anger, arrogance," from the same root as bellows and belly (n.). Meaning "a cause of boasting, occasion of pride" is from 1590s.Related: Boasted; boasting. An Old English word for "boasting" was micelsprecende "big talk."
BOATSWAIN. The warrant officer who in the old Navy was responsible for all the gear that set the ship in motion and all the tackle that kept her at rest. [Sir Geoffrey Callender, "Sea Passages," 1943]
He also summons the hands to their duties with a silver whistle. Phonetic spelling bo'sun/bosun is attested from 1840. Fowler  writes, "The nautical pronunciation (bō'sn) has become so general that to avoid it is more affected than to use it."