How can a CFI renew their certificate with the FAA?
14 CFR 61.197 lists renewal requirements for flight instructors.

If your CFI certificate is still valid (unexpired) than you can renew by any one of the following:


Getting a new rating on your CFI certificate or re-taking a practical test for an existing rating on your CFI certificate


  (1) Passing a practical test for—
  
  (i) One of the ratings listed on the current flight instructor certificate; or
  
  (ii) An additional flight instructor rating; or

Having had at least 5 students in the last 2 years with an 80% or higher pass rate 


  (i) A record of training students showing that, during the preceding 24 calendar months, the flight instructor has endorsed at least 5 students for a practical test for a certificate or rating and at least 80 percent of those students passed that test on the first attempt.

Being a check pilot, chief flight instructor, check airman or flight instructor for a part 121 or 135 operation and active in pilot evaluation during the last 2 years


  (ii) A record showing that, within the preceding 24 calendar months, the flight instructor has served as a company check pilot, chief flight instructor, company check airman, or flight instructor in a part 121 or part 135 operation, or in a position involving the regular evaluation of pilots.

Completing an approved refresher course within the preceding 3 months


  (iii) A graduation certificate showing that, within the preceding 3 calendar months, the person has successfully completed an approved flight instructor refresher course consisting of ground training or flight training, or a combination of both.

Passing a US armed forces military instructor proficiency check in the preceding 12 months.


  (iv) A record showing that, within the preceding 12 months from the month of application, the flight instructor passed an official U.S. Armed Forces military instructor pilot proficiency check.



If you are not actively instructing and just maintaining your certificate the easiest method is the approved refresher course.  There are a few weekend workshops and a online courses that are approved.  The last time I did this method I used an online course and submitted the 8710 via IACRA and completed the whole thing over a weekend and with no physical paperwork.

If your certificate has expired, even if you have completed the above but failed to submit the proper paperwork in time, then you can no longer do this the easy way and have to look at 14 CFR 61.199.  To reinstate an expired certificate you must either pass a practical test for one of your existing CFI ratings or pass a practical test for a new rating for your CFI certificate.