How should a private pilot renew their part 107 certificate?
As a holder of an FAA private pilot certificate, you can take a free course to “renew” a Part 107 certificate. You will first have to create and register an account on the FAA Safety Team website. You should do this anyway to earn WINGS credit toward your next flight review.

The site states..


  The Part 107 Small Unmanned Aircraft Systems (UAS) Recurrent course describes the operational requirements to continue to operate small UAS in the National Airspace System (NAS) under Title 14 of the Code of Federal Regulations (14 CFR) part 107, small Unmanned Aircraft Systems. ***Note: This course may be taken by remote pilots who qualify for relief under Special Federal Aviation Regulation (SFAR) No. 118 to satisfy the aeronautical knowledge recency requirements of 14 CFR 107.65(a) or (b) for a period of six calendar months. FOR PART 61 PILOT CERTIFICATE HOLDERS WITH A CURRENT FLIGHT REVIEW (IN ACCORDANCE WITH 14 CFR PART 61.56), SUCCESSFUL COMPLETION OF THIS ONLINE COURSE SATISFIES THE RECURRENT TRAINING REQUIREMENT TO REMAIN CURRENT AS A PART 107 REMOTE PILOT WITH A SMALL UAS RATING. All other interested individuals may complete this online course as a self-study resource.


A Part 107 certificate does not have an expiration date. You lose the privileges associated with it if you do not renew your currency by taking a refresher course.

The FAR/AIM states...


  §107.65 Aeronautical knowledge recency.
  A person may not operate a small unmanned aircraft system unless that person has completed one of the following, within the previous 24 calendar months:
  
  (a) Passed an initial aeronautical knowledge test covering the areas of knowledge specified in §107.73(a);
  
  (b) Passed a recurrent aeronautical knowledge test covering the areas of knowledge specified in §107.73(b); or
  
  (c) If a person holds a pilot certificate (other than a student pilot certificate) issued under part 61 of this chapter and meets the flight review requirements specified in §§61.56, passed either an initial or recurrent training course covering the areas of knowledge specified in §107.74(a) or (b) in a manner acceptable to the Administrator.