What is the rigorous definition of the verb "to quantize"? I've studied QM and QFT for a couple of years now, so I'm familiar with the tersm "quantize", "quantization" and so on. I'm obviously also familiar with the Lagrangian description of Classical Mechanics. But I'm pretty sure that I lost a piece somewhere in the middle, because every time a classical theory is presented and the book/article/professor goes on saying "and now we quantize the theory" I don't fully understand what is meant.
What is the rigorous definition of the "process of quantization"? Is it just finding the conjugate variables and imposing commutation/anticommutation rules? Or is there more to it?
Bonus question: how does one use the definition of quantization to explain the historical name of QFT, the "second quantization"?
 A: I can share with you what my volume of Merzbacher, 2nd Ed., says on this topic, and this, in turn, may become part of an answer that ultimately satisfies you as you piece this together from what others have to say as well.  We know Hamilton's equations from classical mechanics: $$\dot q_k=\frac{\partial H}{\partial p_k}$$ and  $$\dot p_k=-\frac{\partial H}{\partial q_k}$$
In chapter 15, Quantum Dynamics, Merzbacher derives the time rate of change of an operator: $$i\hbar\frac{d\langle A \rangle }{dt}= \langle AH-HA\rangle$$  or  $$\frac{d\langle A\rangle}{dt}= \frac {\langle AH-HA\rangle}{i\hbar}$$ He then proceeds to apply this to a system of a mass point particle.  He states: "If a quantum system has a classical analog, expectation values of operators behave, in the limit $\hbar \rightarrow0$, like the corresponding classical quantities."
Thus if the operator A above is x (position) or p (momentum), the quantum expression must reduce to the appropriate Hamilton equation above in the limit that $\hbar \rightarrow 0$.
He then writes:

All these conditions can be satisfied if we do the following:
(1) Let H be a Hermitian operator identical in form with $H_c$ [the
classical Hamiltonian], but replacing all coordinates and momenta by
their corresponding operators.
(2) Postulate the fundamental commutation relations between the
Hermitian operators representing coordinates and momenta.

He then proceeds to show that for any two general functions of the coordinates and momenta, F and G, the expression on the RHS of the last equation above is just the Poisson bracket of these two functions from classical mechanics, {F,G} (Note: the quantum expression will have an $\hbar$ factor in it that must be taken to 0 as the correspondence principle requires to complete the equality between the two expressions).
Finally, to come full circle he writes:

In order to test the quantization procedure just outlined, we must show that the quantum mechanics obtained from it are identical with the equations of wave mechanics which we know to give an accurate description of many physical phenomena.

From the title he gives the section, Wave Mechanics Regained, you won't be surprised to learn that he rederives the Schrodinger equation.
As I say, I am just regurgitating what I knew to exist in Merzbacher's QM book.  I hope it contains some piece of the puzzle you are trying to put together!
A: There are quite a number of notions of quantisation. It is a theorem by van Hove & Groenewald that the naive notion of quantisation that is usually taught where position and momentum are mapped to the usual operators cannot hold in all generality, that is the classical Poisson bracket is not mapped perfectly to the commutation relations we would expect.
Other quantisation methods include (and is not limited to) deformation quantisation, geometric quantisation & path integral quantisation.
On a more philosophical view, one might view that the whole process of quantisation is wrong-headed and really we should be looking at classical limits of axiomatically defined quantum systems. That is, the quantum theory comes first, and the classical system is a limit of it.
One example of this view is causal set theory, another example is algebraic  quantum theory where nets of observables, thought of algebras, are posited on spacetime. One outcome here is that the algebra required is theory independent, being a hyperfinite von Neumann factor of type III.
A: There is a section of Leonard Susskind's Quantum Mechanics titled "Quantization."

...We started with a well known and well trusted classical system -the free particle- and quantized it. We codify this procedure as follows:



*

*Start with a classical system. This means a set of coordinates $x$ and momenta $p$... The coordinates and momenta come in pairs, $x_i$ and $p_i$. The classical system also has a Hamiltonian, which is a function of the $x$'s and $p$'s.





*Replace the classical phase space with a linear vector space. In the position representation, the space of states is represented by a wave function $\psi(x)$ that depends on the coordinates -in general, all of them.





*Replace the $x$'s and $p$'s with operators $X_i$ and $P_i$. Each $X_i$ acts on the wavefunction to multiply it by $x_i$. Each $P_i$ acts according to the rule
$P_i \rightarrow -i\hbar\frac{\partial}{\partial x_i}.$




*When these replacements are made, the Hamiltonian becomes an operator that can be used in either the time-dependent or time-independent Schrodinger equation. The time-dependent equation tells us how the wave function changes with time. The time-independent form allows us to find the eigenvectors and eigenvalues of the Hamiltonian.


