The method of appointing people to positions based on rotation rather than on merit.
Unlike the Hobson of Hobson's choice, Buggins wasn't a real person. Buggins is one of the generic names, like John Smith, Joe Blow etc., that were given to the typical man in the street, or as the British used to say, 'the man on the Clapham omnibus'. Incidentally, having been in Clapham recently I noticed (and before the race police start sharpening their pens - I am quite happy with this) the man on the Clapham omnibus is now much more likely to be called Mohammed than Buggins.
A reference to the undistinguished nature of Buggins as a name was printed in The New York Times in August, 1859:
No man likes to be known as Buggins, or Noggs, or Shufflebottom.
There's something of a negative connotation about a position gained by this method - the implication being that, after everyone of merit has had a go, now it's Buggins' turn. Mayors of English towns and cities have long been selected this way. Each year a new mayor is appointed and isn't chosen by the people or on merit but simply by picking the next from the list of the town's notables.
Whenever a name appears in a phrase we always look for a real person.
The name Buggins may have been coined by sailing folk. The first instances of the term 'Buggins turn' in print come from the British admiral John Fisher, 1st Baron Fisher of Kilverstone, who used it more than once in his letters. An example of such a use was printed in A. J. Marder's collection of Fisher's correspondence, Fear God & Dread Nought, 1952. In that publication, Marder reproduced a letter from Lord Fisher, written in 1901:
Favouritism was the secret of our efficiency in the old days. Going by seniority saves so much trouble. 'Buggins's turn' has been our ruin and will be disastrous hereafter!