Correct your error then take steps to prevent it from happening again
This has happened to me before. Our Medicare carrier has a "Clerical Errors Reopening Request Form" so I completed the form with all the information they requested, then asked that they completely delete the claim from their system. Of course, I also filed a new claim for the correct patient, but that is really a separate process from asking Medicare to delete the erroneous claim. Our MAC is Cahaba, but I would think each MAC has a process for correcting clerical mistakes, which is essentially what this sounds like from your description. Check your MAC's website for a similar form, or call the provider service number and ask them how they would prefer you make this correction.
It would also be a good idea to review your processes and procedures for validating patient information before filing claims. The group I work for does not have an EHR, and they use paper superbills. In my practice management system, our patient account numbers are based on the first 3 letters in the patient's last name and the first 3 letters in their first name, so for example, two patients named "Laura Smith" and "Laurel Smith" would potentially have the same account name of SMILAU. If Laura was set-up first, she would be assigned SMILAU, and then when Laurel's account was created, she would be SMILAU00. If there was another patient whose chart name should also be SMILAU, they would be assigned SMILAU01, and so on. As soon as we start adding numbers to the end of a chart name, we flag ALL of the similarly named accounts with an extra reminder to validate the patient's DOB before proceeding. This reminder pops-up in the appointment scheduler for the front desk person, and again on the printed superbill. There is a BOLD message at the top stating NAME ALERT and there's a place for the front desk person to initial that they have validated the DOB. If the superbill has that message and makes it to me in the billing department without initials, it goes back to the office for verification before the claim is filed.
I hope that helps!