When I restart the game, I want a New name other than "Happy". Is there any way to do so?
Option one: Complete the True Pacifist ending, and then select True Reset on the main menu. You will be prompted for a new name.
Option two: Erase your save data (even if you've already done an in-game reset, Undertale still keeps some data around, including the name you chose). This will completely clear out all progress you have made, and make the game "like new."