Why do we use an autoclave at 121°C (250F)? (Origin)
The reason for 121°C and not 120°C is due to how autoclaves work and how they were developed. They sterilize with saturated steam under pressure.  Historically, we measure the pressure generated by the steam. Which leads directly to the answer.
Steam power was the basis of industrial process in the 19th century, and to use it safely required monitoring and controlling boiler pressure. Boiler explosions were not uncommon in the early days of steam power. The Bourdon pressure gauge was invented in 1849 and was a robust technology that yielded reliable and accurate measurement of steam pressure.
https://journals.sagepub.com/doi/pdf/10.1177/002029408702000803
When Chamberland invented the autoclave in 1879 from Papain's earlier pressure cooker, it relied exclusively on pressure measurement for control. Not until 1933 were autoclaves produced that monitored temperature at the drain line to confirm that air was exhausted and the steam was truly saturated.
https://www.steris.com/healthcare/knowledge-center/sterile-processing/everything-about-autoclaves
Study of steam engines was the basis for the development of thermodynamics and the kinetic molecular theory of gasses. Although steam is far from an ideal gas, there is a predictable relationship between temperature and pressure in saturated steam as well as well established properties of heat transfer. These can be found on standard engineering steam tables.
Examples: https://www.engineersedge.com/thermodynamics/steam_tables.htm
Saturated steam is very efficient in transferring heat. This is why scald injuries are so severe compared to simple burns at the same temperatures. It is also how saturated steam kills even heat resistant microorganisms in an autoclave. https://learnaboutgmp.com/good-manufacturing-practices-cgmp/saturated-steam-efficient-sterilization/
One earth atmosphere of pressure at sea level is defined as 14.69 pounds per square inch (psi), though in reality it varies a little bit due to temperature and weather. Because of this variation and because humans using Arabic numerals, arithmetic, and algebra like integer units in 5's; engineers often think of 1 atmosphere of pressure as 15 psi. This amount of saturated steam pressure is well within the capacities of typical steam boilers. This nice round of 15 psi saturated steam will kill even highly resistant bacterial endospores in about 15 minutes. A very reasonable time frame, and a nice mnemonic: 15 psi for 15 minutes.
The temperature of that 15 psi of saturated steam? 121°C.
And that is why we use a very unusual 121°C for typical autoclaving. It's because we are really using a rounded one earth atmosphere of steam pressure. And we did so without concern for direct temperature measurement for the first 50 years of autoclave use. In many ways the temperature of the autoclave was of secondary importance.