"An honest mistake" is something that you did wrong because you didn't know any better.
In the example above, the speaker e-mailed the wrong person because she didn't know that she was supposed to contact the manager first. This is an "honest mistake". If she had already known who to contact, but she sent an e-mail to the wrong person anyway for some tricky reason (like maybe that person usually replies more quickly), that would not be "an honest mistake".
When you call something "an honest mistake", it suggests that people shouldn't get angry at the person who made the mistake. You can use this phrase to reassure someone who's done something wrong:
A: I'm so, so sorry!
B: Don't worry about it. It was an honest mistake.