'Bone dry' clearly derives from an allusion to the dryness of bone after being left in the sun. The equivalent phrase 'as dry as a bone' give testament to that, as virtually all of the 'as X as Y' similes point to a well-known property (i.e. X) of some person or thing (i.e. Y).
The earliest known citation of 'bone dry' is in itself a definition of the phrase in just those terms. It comes in a glossary written by the clergyman Robert Forby in 1830. I've included the full title of Forby's work, as it's always a pleasure to see a title that is longer than the citation taken from it - The Vocabulary Of East Anglia; An Attempt To Record The Vulgar Tongue Of The Twin Sister Counties, Norfolk And Suffolk, As It Existed And Still Exists; With Proof Of Its Antiquity From Etymology And Authority:
"BONE-DRY, adj. perfectly dry; as dry as a bone long bleached in the weather."