Why is stave used instead of chapters in a Christmas Carol?

"Stave" is an older term for a "verse" or "stanza" of a song (or sometimes of a poem). Dickens choose it as a play on the book's title: each section is a "stanza" of the whole "carol".

(Dickens uses the term elsewhere in his writing in precisely this sense of song-stanza. This is not precisely the same as a "musical staff", though it is related to this use of "staff [sometimes "stave" from its plural "staves"] as a term for a set of lines used in musical notation on which to write out a set of notes.)

Note that, after using this device with his first Christmas book (in 1843), Dickens does something similar with the next two: he calls the divisions of The Chimes (1844) "Quarters" after the quarter-hour sounding of clock chimes; The Cricket on the Hearth (1845) is divided into "Chirps".