If you get your neighbor's mail in your mail box and you accidently open it, then realize it isn't yours, and

it is something that you know that person wouldn't want you knowing about (a neighbor), how would you get them their mail in an envelope that is obviously opened?? It is an important peice of mail. The lady and I have the same first name's and I accidently opened it.
Answers:
I would simply retape it, and return it to the mail box. This way the post office will have to remail since it was their mistake for putting it in your mailbox.
If the mail was your exact address, I would retape it. Then I would write on it, "Doesn't exist", which is true because your neighbor doesn't live there. Then the post office will have to return it to the sender.

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Hand it to her face to face and explain the situation.

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Explain, anything else will look like you're trying to hide something or did it on purpose.

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Just be honest about it and apologize, if you try to be sneaky you will get caught eventually and then she won't beleive you when you tell her the truth and will probably think you were just trying to be nosey or steal something.

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i agree wit glen b.

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You hand it to her and say, "I'm so sorry, this was in my mail box and I opened it accidentally, but I stopped reading as soon as I saw that it was your name not mine." Then pretend like the dickens like you never saw anything about the stuff she doesn't want you to know about, and NEVER say a word to anyone about anything you read.

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if you dont want the embarrassment of telling her the truth simply put it in another envelope fill it out to her address with no return address on it and send it to her through the mail...accidents happen ..i have done this myself

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You can give it to your letter carrier with a note stating "opened by mistake." They will deliver it for you, and your neighbor will not know who actually did open it. The post office also has either labels or envelopes they can put it in stating that it was opened by mistake. In the summer there are many casuals, usually students, delivering the mail and these things happen. Nobody is perfect, in any job.

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Tell your neighbor you accidently opened the letter and thought it was from your relative in Hawaii. Apologize profusely and invite them over for barbeque sometime.

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My neighbor opened my electric bill. The outside envelope looks the same for everyone, so it was an honest mistake. He brought it over and apologized. I never thought anything about it.

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walk over at night and put it in her mail box. If it is junk mail toss it.

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I guess I'm just different, I ALWAYS look at the recipient name and address before I open my mail.