This is a family name, which in Ankh-Morpork goes back at least as far, and has a pedigree at least as long and as well-documented, as any of the noble families. It is also the name of the current president of the Thieves' Guild, and is a name proudly borne by many past generations of thieves.
The current Mr Boggis is a slightly sallow, seedy, man in middle age who affects a brown bowler hat, believed to be a badge denoting his professional office. He is not especially bright, and the Gamblers' Guild has noted for professional reasons that he would not be terribly good at poker - his thinking patterns and reactions are far too obvious and he has no concept of subtlety.
Indeed, when 71-hour Ahmed offered twenty camels for Mrs Boggis, he did not realise he was the butt of a joke, and gladly agreed.
Despite being Head Thief, Mr Boggis is also a Special in the Watch. Samuel Vimes tolerates this, as Boggis brings to the Watch his trade expertise as regards hitting people over the head with blunt instruments, while not actually killing them (for to do so would mean the Assassins' Guild calls round for a quiet, but serious, little chat about demarcation). Besides, he comes with two exceedingly large bodyguards who add extra muscle and attitude to a Watch muster.
In The Compleat Ankh-Morpork, the information is given that his full name is Josiah Boggis, and he is no longer a mere "Mr" - he has become Sir Josiah Boggis. No doubt elevating him to the knighthood was another of the Patrician's little intellectual jokes. The forename Josiah is also revealed in A Blink of the Screen. Here we learn that Mr Boggis is all in favour of establishing Boy Scouts in Ankh-Morpork, for young males of a suitable inclination. He believes that the proposed youth organisation would teach many skills not unadjacent to those valued by the Thieves' Guild. he also believes that surviving in the wilderness will be no problem to those who have already survived to double-figure ages in the city.
I offer this without comment. In the British judiciary, there is actually a judge/QC* with the name Judge John Boggis, QC (Recent Judgements)... probably one of those entertaining coincidences...
* Queen's Counsel - a senior advocate.