Why are the quantum numbers $n$ and $\ell$ denoted with those letters? We have 4 quantum numbers: principal, azimuthal, magnetic and spin (denoted $n$, $\ell$, $m$ and $s$ respectively). I assume $m$ and $s$ are simply the initials of 'magnetic' and 'spin'. 
Is there any rationale behind the $n$ and $\ell$ denomination? 
 A: The principle quantum number is denoted $n$ because it is a 'natural number', $n=1,2,3,4....$.
The secondary azimuthal quantum number (also known as orbital angular momentum) is denoted $\ell$ through its association with angular momentum (typically denoted $L$).
A: You are right to think that $4$ quantum numbers completely describe the state of an electron within an atom. I'll explain each in turn.
$n$ is related to the energy of the electron, i.e. the eigenvalue of the Hamiltonian operator. It's called the principal quantum number since we usually think of the energy as the most basic quantum quantity. We use the letter $n$ because we usually take $n$ to be a natural number. For example in the case of hydrogen
$$E_n =-13.6eV/n^2$$
$l$ is related to the total orbital angular momentum, i.e. the eigenvalue of the angular momentum operator $\mathbf{L}^2$. It's known as azimuthal roughly because angular momentum is associated with rotational symmetry. I'm not aware of a good reason why we denote angular momentum by the letter $L$, although a quick Google gives you several options.
$m$ is related to the $z$ component of orbital angular momentum, i.e. the eigenvalue of the angular momentum operator $L_z$. It's called the magnetic quantum number because it determines how a (spinless) atomic orbital changes in the presence of a magnetic field. 
$s$ is the spin of the electron, which can be up or down. 
The first three quantum numbers are interrelate, because solving the Schrodinger equation places constraints on which $l$ and $m$ are allowed for each $n$.
For more details, see this Wiki article!
