日 is sun and 本 is origin (in this case), which lines up with the western moniker for Japan as the land of the rising sun.

Also I’d say that goodbye is a decent enough translation for sayonara. I can’t remember the last time I said goodbye in English (instead of bye or see you), and seems almost as formal and final as farewell, which seems almost outdated at this point. I’m not sure I’ve ever actually used that in my entire life.