The day of the interview you can tell them what name you want to use. They will be curious as to why you have been using a green card with the wrong name for years. Be ready to explain or better yet, talk to an immigration now about the issue.
You really ought to have brought this to their attention sooner. However, Naturalization presents an easy opportunity to correct the error.
IMPORTANT: Mr. Murray's response is NOT legal advice and does NOT create an attorney-client relationship. You should NOT rely on this response. Mr. Murray's response was generated without conducting a full inquiry as would occur during an attorney-client consultation. It is likely that the response above may be made less accurate, or become entirely inaccurate, as you, i.e. the questioner, disclose additional facts that should only be discussed during a private attorney-client consultation. I strongly recommend that you consult an attorney who is licensed to practice law in your state (or, in the case of immigration law, an attorney in ANY state), whereupon all relevant facts will be discussed. All responses posted by Mr. Murray on Avvo.com are intended as general information for the education of the public, and not for any specific individual. For persons located in New Jersey: To the extent that Mr. Murray's profile can be considered an advertisement in New Jersey, which is denied, be advised that NO ASPECT OF THIS ADVERTISEMENT HAS BEEN APPROVED BY THE SUPREME COURT OF NEW JERSEY. Furthermore, the selection methodology for the SuperLawyers' "Rising Stars" awards is set forth at length at this website: http://www.superlawyers.com/about/selection_process.html.
USCIS will let you change your name in the naturalization process.
The answer provided here is general in nature and does not take into account other factors that may need to be reviewed for a more precise answer. You should consult with an immigration attorney before taking any action. The answer here is not intended to create an attorney-client relationship.
There are two ways to fix your name: a name correction with evidence of the correct name OR a legal name change once you are sworn-in as a US Citizen. if the "slightly wrong" name on your green card is a misspelling or in the wrong order and you have proof of the correct spelling or name order (birth certificate, school records, etc.), then it is a name correction and just bring your evidence to your interview and explain to the officer. If the name on your green card is "slightly wrong" just as a matter of preference, then you would need a legal name change. All of this should be discussed at your interview. If you live in an area where all the ceremonies are at the Federal Court, then it does not matter which route you go and the name change would probably just be easier. If you live in an area where ceremonies are scheduled at the Federal Court and other places, then going the route of the legal name change might delay your swearing-in ceremony because you must be sworn-in before a judge when you are doing a legal name change and you would have to wait for a spot in a ceremony at the Federal Court.
Either way, do not worry to much about this. A lot can be explained/corrected at interview. Good luck!