The way the token's use is described, First Player is probably the best name for it.
Unlike Battlestar Galactica, for example, where the Current Player token is passed at the end of each players turn, in Age of Conan, the rules state that the first player token is only passed when gameplay is interrupted to serve as a reminder of who the first player will be once gameplay resumes.
Here is the rule on page 10:
If gameplay is ever interrupted due to the end of an adventure or an age change phase, the player would take the next turn takes the first player token. Once gameplay resumes, the player with the first player token takes the first turn and gameplay continues as normal.