Mail fraud is a serious crime. Do everything the other posters said. I'd start by taking it to the PO and going right to the counter. The clerk may handle it or may refer you to a fraud person or the postmaster. Explain that it's not an error by the sender, that the neighbor wrote her name in your box.
If you just send it back to the sender, it doesn't get dealt with as fraud. You can ask the PO what to do about things in the future.
I still get stuff for the former owner of this house (only been gone 25 years!) but that's not fraud, and I have returned it and/or written letters of explanation (for example, to his university which sends all kinds of alumni information). Now I just throw it out since they choose not to get the message or to notify all departments.
If I get mis-addressed junk mail, I toss it. If I get something for a mystery individual, I write "Addressee unknown" and put it in the outgoing mail.
But your situation is more serious than this and I'd handle it officially. And ask the PO what to do about things from state agencies or any government outlet - whether you should handle it or keep giving it to the Post Office. I would not handle it directly with your letter carrier as that puts all the pressure on that person and your neighbor could take it out on her/him.
And keep refusing her rides!