You can say it either way. An accident is something that you don't expect to happen, and don't mean to do. A mistake is an error. So hitting the wrong button was a mistake, as well as an accident.

You can use either phrase.

As far as I can tell, they're the same thing. A very subtle difference is that an 'accident' is unintentional, but a 'mistake' is not necessarily (it could also have been a poor, but intentional, decision). Here, 'by mistake' clearly means 'accidentally'. It's a bit more natural to say "It was an accident" in this case, in my opinion.