| This is a phrase which I have heard used in Australia. It can have several different meanings.
Firstly, it has a sexual connotation. To tell someone to give the dog a bone means to tell them to have sex with someone; to satisfy the third person's sexual desire. In this case the phrase is somewhat derogatory, and usually refers to a male 'giving' to either a female or a male.
Secondly, the phrase can mean to give someone a second chance at something. For example, a colleague at work might ask you to "give the dog a bone", signifying that you should give one of your employees a second chance at his job, instead of firing him right away.
I guess you could also use it in its literal meaning - to feed your dog with a bone. Dogs like to munch on bones for the purposes of keeping their teeth sharp, and also for getting the leftover flesh and bone marrow.
Lastly, the phrase appears in the children's nursery rhyme 'This Old Man', which my mother used to sing to me when I was younger. | 
