There is possibility to change menu name dynamically, menus are saved in the database which means that you will have to always modify the  menu record in the database.

Expected issues

When you have multiple user login how are you going to manage the name change, you have only one menu to rename for multiple users.

Except if you can hunt down the menu using JavaScript customization I will advice not to do that, you can find another way around by filtering records based on the current login user using record rules.

A clever way of naming menu like My Orders, My Expenses the apply a domain on the action that open up the menu will serve the same purpose.