I've never heard of it in relation to midshipmen. I looked in Moore's British Mariner's Vocabulary from 1801. The only keeper listed is "boat keeper," which is self-explanatory.
Thanks for looking Susan. When I first read the passage, I failed to connect the two volunteers to the keeper midshipman entirely. My first thought was "time keeper". Once Mary made the connection, I re-read the whole section several times. Nearby, Stockwin has the captain say that the two "volunteers first class" had been rated as "captain's servants" making them "apprentice midshipmen".
All that explanation makes me think that Mary's opinion is in line with Stockwin's intention. Apprentice midshipmen would certainly need an experienced midshipman as a "sea-daddy".
Whether "keeper" is a valid AoS term is the question. Since Stockwin said that the two volunteers had been placed in the charge of the midshipman, using the term "keeper" was certainly not needed, so why was it used at all? Where did Stockwin get "keeper"? Is it just a novelist's license?