If a person shows interest in a name, it means they are curious about it for some reason. There are several reasons a person would be interested or curious in a name. First you must identify what the "Name" pertains to. A name of a book? A name of a movie? Those would also be called Titles. If a person is interested in a name by what a person is called, it could mean that they would like to find out the meaning for a personal reason. Perhaps they are going to have a child and would like to find a good name for it. If they are interested in the meaning of a name, they could tie that meaning of the person they are referring to.