The story
Earliest recorded use is by Admiral Fisher, later First Sea Lord, in a 1901 letter. 'Buggins' is a humorous invented surname, suggestive of a man of mediocre ability.
Etymology from Wiktionary, CC BY-SA 4.0.
Buggins's turn
Today it means: A humorous, disparaging term for the appointment of a person by rotation or promotion, on the basis of length of service (it being that person's "turn") rather than merit or level of qualification. Full definition →