The etymology of euthanasia is pretty straightforward, as this site shows:
Early 17th century (in the sense ‘easy death’): from Greek, from eu ‘well’ + thanatos ‘death’.
What is less straightforward, though, is how come -thanasia was formed off the Greek noun thanatos.
In my opinion, -thanasia should have been the end result if the Greek word for "death" would have been
*thanasos: < *thanas- + -ia.
Hence, based on the same model, euthanasia should really have been
*euthanatia: *thanatos: < *thanat- + -ia.