Something that two or more people argue about strongly over a long period of time. The commonly accepted origin of the phrase is the image of two animals, usually dogs, fighting over the last bone of dinner. Neither will give up his position, and the bone is seen as the specific point of arguing.
A similar phrase bone of dissension was used in the sixteenth and seventeenth centuries, but fell out of favor. Either the phrase morphed into the current bone of contention or contention evolved freely and outpaced the other in popularity.
Example
The main bone of contention between us is our children’s education.
Low attendance has become a bone of contention between students and the management of a city college
Every day there is a new bone of contention, which is cause for some bitter argument, some that cross all lines while taking on the opponent.