Ask eDreams to cancel the tickets? You are still within 24 hours of booking - but admittedly, I don't know if eDreams or Alitialia have a 24-hour grace period/24 hour cancellation rule, but you can ask.

Anyway, if they can't fix it and the only solution offered is to buy a new ticket - I guess you will have to chance it at check-in - I wouldn't cancel/buy new ticket but take the chance. There is no requirement that this name error is fixed; as long as you are able to check-in and get a boarding pass, you should be fine. The risk is obviously, that the check-in agent will refuse to issue boarding pass because first name in passport and reservation do not match. Whether that risk is high or low I have no idea - much will depend on the check-in agent that you approach. Some check-in agent seem to be able to fix these issues at check-in, but again, it all depends on the person that you talk to.

As you may have found out by now, booking air tickets via an OTA like eDreams certainly doesn't help you when things go South as in your situation. It becomes far more complicated with 3 parties involved (and 2 of them pointing at each other)

Good luck!