From what I've read, the favourable opinion is that it's named after Vespucci. However, on an episode of QI I've just watched they rejected this and claimed it is actually named after Amerike.
hi! there's room for more info on this, but you can catch up with previous responses here
When naming America, why do they use Amerigo Vespucci's first name, as opposed to his last?
Why was the New World named after Amerigo Vespucci rather than Christopher Columbus?
Why was America named after Amerigo Vespucci? - includes some incidental discussion on Richard Ameryk/Amerike/ap Meurig