I think Ms. Jones trusted Roger for a number of reasons. Firstly, he had been apologetic and remorseful for snatching her purse and she believed his sentiments to be sincere. She had also told him that if he needed something all he needed to do was ask as opposed to stealing. She was therefore confident that by leaving her purse within his vicinity and the door wide open, he would not have attempted anything. In addition to that, she empathized with him for having to fend for himself as well as lacking someone to guide and counsel him. Her empathy was also heightened by the fact that he reminded her of herself during her youth when she had to do all kinds of things in order to get the things that she wanted but could not afford. In a nutshell, I think she trusted Roger because he showed remorse towards his actions and she empathized with him.
I think Ms Jones Trust Roger because he was just a little kid and she new that if she takes him to her house she will teach him right and wrong