You don't even need an Arduino for that.
- Get a bluetooth module (such as JY-MCU HC-05)
- Plug the module to your computer using a USB-to-UART bridge (such as a FTDI breakout board)
- Using a serial terminal program (such as Arduino Serial Monitor, Windows Terminal or Putty) connect serially to the BT module and configure it to the name you want (using AT+ commands) and a password.
- disconnect from the computer
- power the module using anything from 3.6 to 5V.
That's it. The module will broadcast itself with the name you configured in the step above. If you configure a really hard password, nobody will be able to connect to it anyway.