Why was Carbon-12 chosen for the atomic mass unit?
As is the case elsewhere in metrology, the answer is tied up in history, measurability, practicality, repeatability, past misconceptions, and consistency (despite those past misconceptions).
The history of atomic mass and the mole (the two are quite interconnected) goes back to the early 19th century to John Dalton, the father of atomic theory. The unified atomic mass unit is named after him. Scientists of that era were just learning about elements; the periodic table was 60 years in Dalton's future. Dalton initially proposed using hydrogen as the basis. Issues of measurability and repeatability quickly cropped up. So did mistakes. Dalton, for example, thought water was HO rather than H2O.
These issues resulted in chemists switching to to an oxygen-based standard based on the oxygen found on Earth. (That elements can come in multiple isotopes was not known at this time.) Physicists' investigations at the atomic level caused them to develop their own standard in the 20th century, based on 16O rather than the natural mix of O16O, 17O, and 18O (atomic masses: 15.994915, 16.999131, and 17.999161, respectively, with a nominal mix of 379.9 ppm for 17O, 2005.20 ppm for 18O, and the remainder 16O) used by chemists.
The natural mix of the various isotopes of oxygen is not constant. It varies with time, place, and climate. Improved measurements and more widespread usage made repeatability become a significant issue by the middle of 20th century. The primary cause is natural variations in the two most common isotopes of oxygen, 16O (the dominant isotope) and 18O (about 2000 parts per million, on average). The IUPAC Technical Report on atomic weights of the elements lists the atomic weight of naturally occurring oxygen as varying from 15.99903 to 15.99977.
The primary cause of these natural variations is the preferential evaporation and precipitation of water molecules based on various isotopes of oxygen. Water based on 16O evaporates more slightly readily than does water based on 18O, making tropical oceans a bit concentrated in 18O compared to average. On the flip side, water based on 18O precipitates slightly more readily than does water based on 16O. This makes precipitation in the tropics have slightly higher 18O concentrations compared to nominal, and it makes precipitation in high latitudes have slightly lower 18O concentrations compared to nominal.
Physicists had a solution: Switch to their isotopically pure 16O standard. This would have represented an unacceptably large change (275 ppm) in chemistry's oxygen-based standard. It would have required textbooks, reference books, and perhaps most importantly, the recipes used at refineries and other chemical factories to have been rewritten. The commercial costs would have been immense. It's important to keep in kind that metrology exists first and foremost to support commerce. Chemists therefore balked at that suggestion made by physicists.
The carbon-based standard represented a nice compromise. By chance, defining the atomic mass as 1/16th of the mass of a mole of oxygen comprising a natural mix of O16O, 17O, and 18O is very close to a standard defining the atomic mass as 1/12 the mass of a mole of 12C . This represented a 42 ppm change from the chemists natural oxygen standard as compared to the 275 ppm change that would have resulted from changing to 1/16 of the mass of a mole of 16O . This new standard was based on a pure isotope, thereby keeping physicists happy, and it represented an acceptably small departure from the past, thereby keeping chemists and commerce happy.
Britannica.com on John-Dalton/Atomic-theory entry
I'm leary of referencing wikipedia. Britannica is still fair game for basic facts.
Class 11: How Atoms Combine
Dalton's mistake on assuming water was diatomic is widely reported. This is one of many sites that make this claim on Dalton's mistake.
Holden, Norman E. "Atomic weights and the international committee–a historical review." Chemistry International 26.1 (2004): 4-7.
I found this after the fact, after Emilio Pisanty asked me to find some references. This says everything I wrote, only better, in more detail, and with lots of references.
Meija, Juris, et al. "Atomic weights of the elements 2013 (IUPAC Technical Report)." Pure and Applied Chemistry 88.3 (2016): 265-291.
See table 1, and also figure 6.