Old English ban "bone, tusk, hard animal tissue forming the substance of the skeleton; one of the parts which make up the skeleton," from Proto-Germanic *bainan (source also of Old Frisian and Old Saxon ben, Old Norse bein, Danish ben, German Bein). Absent in Gothic, with no cognates outside Germanic (the common PIE root is *ost-); the Norse, Dutch, and German cognates also mean "shank of the leg," and this is the main meaning in Modern German, but English seems never to have had this sense.
To work (one's) fingers to the bone is from 1809. To have a bone to pick (1560s) is an image from dogs struggling to crack or gnaw a bone (to pick a bone "strip a bone by picking or gnawing" is attested from late 15c.); bone of contention (1560s) is from two dogs fighting over a bone; the images seem to have become somewhat merged. Also compare bones.
Bone-china, which is mixed with bone-dust, is by 1854. Bone-shaker (1874) was an old name for the early type of bicycle, before the adoption of rubber tires, etc.
especially in bone up "study," 1880s student slang, probably from "Bohn's Classical Library," a popular series in higher education published by German-born English publisher Henry George Bohn (1796-1884) as part of a broad series of "libraries" he issued from 1846, totaling 766 volumes, continued after 1864 by G. Bell & Sons. The other guess is that it is an allusion to knuckle-bones and has the same figurative sense as the verbal phrase knuckle down "get to work."