The term 'his nibs' is a mock title used to refer to a self-important man, especially one in authority. It is modelled after the pattern of references to the British aristocracy, such as his Lordship.
The origins of the term are obscure, with such sources as the acclaimed Oxford English Dictionary simply listing it alongside the words 'source unknown'; it is first recorded in print in around 1820, although it can be reasonably assumed that it was in common use much earlier than the given date. There is some evidence that nibs is a variant form of nabs, and that both may have their origin in the ancient word neb, meaning a beak or nose. Webster lists 'nib' as a general synonym for 'neb'; the word nib itself was once used as a slang term for a gentleman, as was yet another word shifted in vowel sound which can still be heard today: nob. It seems the vowel itself was highly fluid, which is hardly surprising given the variety of dialects the term has passed through over time.
An interesting extension of the etymology of his nibs is the connection to those who consider themselves so superior that they 'have their noses (nebs) in the air'.