'Buggins' turn' refers to the practice of assigning appointments to persons in rotation, rather than on merit. The OED records this and gives examples of its use from 1901.
As regards etymology it just says 'typical proper name used generically'. But why Buggins? Does anyone know who the original Buggins was? Or was it a name that was earlier used in the way Joe Bloggs is today? Sounds the sort of surname Dickens might have created but he is not found in a list of characters.
Simon Lamb (below) may have nearly got there with his answer; Lady Cecilia's first father-in-law was one Barrington Buggin, whom Brian Dietz speculated was the original subject of the expression. In the mid-eighteenth century Buggin controlled more than half of London's Legal Quays, and as leader of the wharfingers was succeeded by his son of the same name in 1780 - because (it was noted) "Barrington Buggins stands next in turn".
According to the following source it appears that there is no reference to a real person:
'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:'
'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:'
Buggins appears in Anthony Trollope's Barchester Chronicles. He was a civil servant who was leaving his job, and a large number of possible candidates were pursuing the position. Pre-dates all other suggestions - i.e. 19th century.
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Editing to add a reference to this potentially intriguing answer.
"Now, would you believe it? I have used up three lifts of note-paper already in telling people that there is no vacancy for a lobby messenger in the Petty Bag office. Seven peeresses have asked for it for their favourite footmen. . . ."
Buggins was the messenger for whose not vacant place all the peeresses were striving with so much animation.
(Anthony Trollope, Framley Parsonage, Ch. XVIII. Originally published 1860, per Wikipedia.)
2nd wife of the Duke of Sussex (6th son of George Third) was Lady Cecilia Buggin. They married in 1831. His 1st marriage was annulled because he had not sought the permission of his father, the King!
