Is there a way to rename the built in speaker name? (Mac Mini M1)

You can't rename the existing audio output, but you can create another one with your chosen name.

Use Audio MIDI Setup.app (in the Utilities folder) to create an Aggregate Device that only contains your mini's speakers. (Normally, this is used to create a single input/output for multiple devices.)

Click on the + button at the bottom of the window to create a new Aggregate device, and then double-click on the item in the sidebar to rename it.

You will need to tick the "Use" checkbox next to "Mac mini Speakers".

You can then choose a more descriptive name, in preference to "Mac Mini Speakers".

This will be listed in System Preferences, in addition to the built-in audio output names. (However, it seems that the menulet doesn't get updated immediately. You'll need to log in again.)