Use the noun namesake to describe the recipient of a handed-down name, like Bob Jr., or Ricky Smith III. Less often, the word also means anyone who shares a name with someone else, so you could refer to all the Emmas in your school as namesakes. The first recorded use of the word namesake was in the mid-1600s, and it probably began as the phrase "for the name's sake," before being condensed into a single word.