Why Pauli called the swap matrix $σ_x$? Why not $σ_y$? Why Pauli called the following matrix $\:\sigma_x\:$ and not $\:\sigma_y$? 
\begin{equation}
\sigma_x\boldsymbol{=}
\begin{bmatrix}
0 & 1 \vphantom{\tfrac{a}{b}}\\
1 & 0 \vphantom{\tfrac{a}{b}}
\end{bmatrix} 
\tag{01}\label{01}
\end{equation}
 A: Any vector in  $\mathbb{R}^3$ can be represented by a $2\times2$ hermitian traceless matrix and vice versa. So, there exists a bijection (one-to-one and onto correspondence) between $\mathbb{R}^3$ and the space of  $2\times2$  hermitian traceless matrices, let it be $\mathbb{H}$ :
\begin{equation}
   \mathbf{r}\boldsymbol{=}(x,y,z)\in \mathbb{R}^3\;\boldsymbol{\longleftrightarrow} \; 
   \mathrm R= 
   \begin{bmatrix}         
             z         &   x\boldsymbol{-}iy  \\
           x\boldsymbol{+}iy   &       \boldsymbol{-}z
   \end{bmatrix}
       \in \mathbb{H}      
 \tag{01}
\end{equation}
From the usual basis of  $\mathbb{R}^3$
\begin{equation}  
 \mathbf{e}_x\boldsymbol{=}\left(1,0,0\right),\quad 
 \mathbf{e}_y\boldsymbol{=}\left(0,1,0\right),\quad \mathbf{e}_z\boldsymbol{=}\left(0,0,1\right)
  \tag{02}
\end{equation}
we construct  a basis for $\mathbb{H}$
\begin{align}
   \mathbf{e}_x & \boldsymbol{=}(1,0,0)\qquad \boldsymbol{\longleftrightarrow} \qquad \sigma_x\boldsymbol{=} 
   \begin{bmatrix}         
         \:\: 0 & \hphantom{\boldsymbol{-}}1\:\:\vphantom{\dfrac{a}{b}}\\
         \:\: 1 & \hphantom{\boldsymbol{-}}0\:\:\vphantom{\dfrac{a}{b}}
   \end{bmatrix}
  \tag{03a}\\
   \mathbf{e}_y & \boldsymbol{=}(0,1,0)\qquad \boldsymbol{\longleftrightarrow} \qquad \sigma_y\boldsymbol{=} 
   \begin{bmatrix}         
         \:\: 0 & \boldsymbol{-}i\:\:\vphantom{\dfrac{a}{b}}\\
         \:\: i & \hphantom{\boldsymbol{-}}0\:\:\vphantom{\dfrac{a}{b}}
   \end{bmatrix}
  \tag{03b}\\
   \mathbf{e}_z & \boldsymbol{=}(0,0,1)\qquad \boldsymbol{\longleftrightarrow} \qquad \sigma_z\boldsymbol{=}
    \begin{bmatrix}         
         \:\: 1 & \hphantom{\boldsymbol{-}}0\:\:\vphantom{\dfrac{a}{b}}\\
         \:\: 0 & \boldsymbol{-}1\:\:\vphantom{\dfrac{a}{b}}
   \end{bmatrix}    
  \tag{03c}
\end{align}
where $\:\boldsymbol{\sigma}\equiv(\sigma_x,\sigma_y,\sigma_z)\:$ the Pauli matrices.
Note also that the  matrix
\begin{equation}  
 U\boldsymbol{=}\cos\tfrac{\theta}{2}\,\mathrm I\boldsymbol{-}i\sigma_x\sin\tfrac{\theta}{2}
 \boldsymbol{=}
 \begin{bmatrix}         
         \:\: \cos\tfrac{\theta}{2} & \boldsymbol{-}i\sin\tfrac{\theta}{2}\:\:\vphantom{\dfrac{a}{b}}\\
         \:\: \boldsymbol{-}i\sin\tfrac{\theta}{2} & \hphantom{\boldsymbol{-}} \cos\tfrac{\theta}{2}\:\:\vphantom{\dfrac{a}{b}}
   \end{bmatrix}
\tag{04}
\end{equation}
is the unitary matrix representation of the rotation around the $x$-axis through an angle $\theta$.
A: The matrix 
$$
\sigma_x=\left(\begin{array}{cc} 0&1 \\ 1&0\end{array}\right)
$$
is just the (standard) permutation matrix interchanging two objects: if
$$
\vert 1\rangle \to \left(\begin{array}{c} 1 \\ 0\end{array}\right)\, ,\qquad 
\vert 2\rangle \to \left(\begin{array}{c} 0 \\ 1\end{array}\right)\, ,
$$
then 
$\sigma_x\vert 1\rangle =\vert 2\rangle$ and $\sigma_x\vert 2\rangle=\vert 1\rangle$ without any phases. $\sigma_y$ introduces an additional phase to the permutation operation.
