If you want to make a MIDI USB device, I would advise you to use a Teensy as it is really well integrated. Teensy still today stay my first choice when I want to make a USB MIDI device. If you want to change your MIDI device name with a Teensy, you have an exemple sketch in the IDE and explanations here.
That said, you can use an Arduino with MIDIUSB library. Note that this library is only working with atmega32u4 based boards and ARM boards, so, that would mean Leonardo, Micro, LilyPad, Esplora, Due, Zero, YŚN, Beetle Arduino, Arduino pro micro (this one has lots of variants, with or without USB connector), A-Star 32U4 Micro (Confirmed that is worked by Sam Kusnetz, thank you Sam!), Arduino Pico, Nerdonic Exen Mini, Adafruit Feather 32u4, Itsy Bitsy 32u4, Atmega32u4 Breakout Board, Adafruit Bluefruit LE Micro, Arduboy, Flora, Circuit Playground Classic, Bare Conductive Touch Board, Mojo v3, Fio v3, Qduino Mini, Dreamer Nano V4, LeoStick, WIOT Board, Papilio DUO, µduino, Duinobot, Tah, IMUduino, mCookied, Neutrino, to name a good few that might work! If you think of another, let me know!
Once you made your MIDI USB device, you will probably want to give it another name than the one automatically given when compiling. Here is how to do it. Here I have tested it for the Leonardo and it would be easy to adapt to another Arduino compatible board.
leonardomusinou.build.vid=0x2341 leonardomusinou.build.pid=0x8031 leonardomusinou.build.usb_product="Arduino Leonardo musinou"
Well, if you have another board or if you have to change the name multiple times, or if there is a compatibility problem your your IDE version, here is what you have to do