E. Cobham Brewer 18101897. Dictionary of Phrase and Fable. 1898.
Bone to pick (A).
A sop to Cerbêrus. A lucrative appointment given to a troublesome opponent in order to silence him. Thus Chisholm Anstey was sent to Hong-Kong as a judge to keep him away from the House of Commons. Of course the allusion is to throwing a bone to a dog barking at you.
In those days the usual plan to get rid of an oratorical patriot in the House was to give him a bone to pick.Anthony Collins.
I have a bone to pick with you. An unpleasant matter to settle with you. At the marriage banquets of the Sicilian poor, the brides father, after the meal, used to hand the bridegroom a bone, saying, Pick this bone, for you have taken in hand a much harder task.