Name of concept: Replace classical variables by quantum operators

Canonical quantisation was introduced by Dirac in 1926

Canonical quantization mentioned by @CharlesFrancis is probably the most precise answer. Colloquially one would often use term correspondence principle, also known as Bohr's correspondence principle, and the Ehrenfest theorem. However in reality these have more to do with the system's behavior than with the actual quantization procedure.

Note in this context that some phenomena, such as spin, have no classical analogs.