The words 'bone' or 'bones' have often been used to imply the very essence of someone/something. You find this in sayings such as: "I feel it in my very bones", "Frozen to the bone" and so on. Accordingly, 'bone idle' just means idle through and through, idle to the marrow or idle in very essence.
I should have added above that the earliest recorded written use of the phrase appeared in a letter from the historian, Thomas Carlyle in the 1830s. ('Bone lazy' had appeared somewhat earlier.) So far as we know, therefore, the Carlyle use is the origin.