I think that you are talking about what happens in Chapter 4. At the end of Chapter 3, the boarder, Billy Bones, dies. In Chapter 4, Jim and his mother believe that they ought to have at least part of what is in the chest.
The reason that they believe this is that Billy Bones had been living at their inn for quite a while now and he owed them money. So they believed that they deserved his money as much as his old shipmates did. Neither of them actually exactly says this, but Jim refers to Bones's debts and his mother says the money is lawfully theirs.