I found this to be a confusing process. There is no secondary information except scattered ValueMD posts.
I will tell you how to do this and what to expect along the way. It can take up to a week to receive your revised permit.
This ECFMG link has some information. Review the table with application processing deadlines.
1. Cancel your exam with Prometric
If it is more than 31 days before your exam, there is no fee to pay, proceed to the next step.
If it is less than or equal to 31 days before your exam, you need to pay a fee to Prometric, call 800-633-3926. [Update 6/25/2014: You may not need to pay the Prometric cancellation fee to extend your eligibility anymore, please see this comment from another student.]
At the time of this writing there is no telephone prompt for reinstating eligibility. Follow the prompt for scheduling and other options. You will end at a prompt asking for your scheduling number. If it is past your eligibility period your number may be denied. If this happens, dial 0#. You may have try this up to four times before the system transfers you.
Someone will come on the phone. Tell them you need to extend your eligibility period for the USMLE Step 1 and it is less than 31 days before your scheduled exam date. They will ask a few security questions and transfer you to someone else. The next person will process your payment.
You will receive a receipt by email. This needs to be forwarded to firstname.lastname@example.org. Include your permit number at the top of the forwarded message.
Four hours later I received a reply. The subject was USMLE Step 1 Scheduling. My ID number was at the top of the message. It confirmed I paid the appointment change fee and Prometric reset my registration. After the ECFMG processes my extention request they will update my Prometric record and email my revised scheduling permit.
2. Send the application
If you are a US or Canadian medical graduate your point of contact is the NBME. Login here. Click on the link to submit a new application. It will take you to the requirements page. At the bottom there is a link to a form which must be mailed.
Foreign medical graduates must login to ECFMG online services. The first page has a link for the application to extend eligibility. There is a $65 fee at the end of the online application.
You only need to resubmit form 186 if it has been more than a few years since you applied. The end of the online application will tell you if you need to submit your certification of identification.
My Post-Application Timeline
Update 0 / 4 hours: ECFMG confirms payment was made to Prometric.
Update 1 / 12 hours: “Your Eligibility Period Extension (EPEx) request has been submitted to ECFMG. You will receive email confirmation when your request has been processed.”
Update 2 / 1 day, 6 hours: My eligibility period has been extended to (date). The NBME will issue a new permit and the ECFMG will email it to me in about one week.
Update 3 / 3 days, 12 hours: ECFMG emailed new permit.
Applied Monday morning and received new permit Thursday night.
I wish you the best for your new eligibility period. Remember, a mistake is only a problem if you don’t learn from it. It is important to take your exam during your extended eligibility period.
Please leave all questions in the comments. Do not email me directly. I will respond here and send a copy to your email. If you have a follow up question, please ask it in the comments.