How insert a text on middle right of eqnarray

Don't use eqnarray or eqnarray*. See the posting eqnarray vs align for an in-depth discussion of this issue.

Use one of the multi-line equation environments of the amsmath package instead. For the case at hand, I suggest you employ a split environment. The amsmath package provides the instruction \tag; it would appear to meet your typesetting requirement perfectly.

While you're at it, you may also want to increase the sizes of the round parentheses and curly braces in the second row.

\documentclass{article}
\usepackage[letterpaper,margin=1in]{geometry} % set suitable page parameters
\usepackage{amsmath} % for 'split' env. and '\tag' macro
\begin{document}

\[
\begin{split}
\tau_{h}^{*}(x^{*}) 
&= \frac{-(1 - \lambda V^{*\infty}\tau_{h,in}^{*})x^{*} 
  + (1 + V^{*\infty}\tau_{h,in}^{*})}{(1 + \lambda)V^{*\infty}} \\
\tau_{c}^{*}(x^{*}) 
&= \Bigl(\frac{1}{1 + \gamma}\Bigr)
   \Bigl\{1 - \gamma \tau_{h,in}^{*} 
      - \frac{\gamma}{(1 + \lambda) V^{*\infty}}\Bigr\} x^{*} 
+ \Bigl(\frac{1}{1 + \lambda}\Bigr)
  \Bigl(\frac{1}{V^{*\infty}} + \tau_{h,in}^{*}\Bigr) 
\end{split}
\tag{First case}
\]
\end{document}

Here are three possibilities, one is based on the empheq package, which loads mathtools, which loads amsmath. I also improved a bit the look of the equations, removing unnecessary parentheses.

    \documentclass{article}
    \usepackage[showframe]{geometry}
    \usepackage{empheq, amssymb}

    \begin{document}

    \begin{alignat*}{2}
        % \nonumber % Remove numbering (before each equation)
          \tau_{h}^{*}(x^{*}) &=\frac{-(1 - \lambda V^{*\infty}\tau_{h,in}^{*})x^{*} + (1 + V^{*\infty}\tau_{h,in}^{*})}{(1 + \lambda)V^{*\infty}} & & \\
    \ArrowBetweenLines*[\text{First case}]
          \tau_{c}^{*}(x^{*}) &=\frac{1}{1 + \gamma}\Bigl(1 - \gamma \tau_{h,in}^{*} - \frac{\gamma}{(1 + \lambda) V^{*\infty}}\Bigr) x^{*} + \frac{1}{1 + \lambda} \Bigl(\frac{1}{V^{*\infty}} + \tau_{h,in}^{*}\Bigr) & &
        \end{alignat*}
    \bigskip

    \begin{empheq}[right=\text{\quad First case}]{align*}
          \tau_{h}^{*}(x^{*}) &=\frac{-(1 - \lambda V^{*\infty}\tau_{h,in}^{*})x^{*} + (1 + V^{*\infty}\tau_{h,in}^{*})}{(1 + \lambda)V^{*\infty}} \\[1ex]
          \tau_{c}^{*}(x^{*}) &=\frac{1}{1 + \gamma}\Bigl(1 - \gamma \tau_{h,in}^{*} - \frac{\gamma}{(1 + \lambda) V^{*\infty}}\Bigr) x^{*} + \frac{1}{1 + \lambda} \Bigl(\frac{1}{V^{*\infty}} + \tau_{h,in}^{*}\Bigr)
    \end{empheq}
    \bigskip

    \begin{equation}\tag*{First case}
      \begin{aligned}
          \tau_{h}^{*}(x^{*}) &=\frac{-(1 - \lambda V^{*\infty}\tau_{h,in}^{*})x^{*} + (1 + V^{*\infty}\tau_{h,in}^{*})}{(1 + \lambda)V^{*\infty}} \\[1ex]
          \tau_{c}^{*}(x^{*}) &=\frac{1}{1 + \gamma}\Bigl(1 - \gamma \tau_{h,in}^{*} - \frac{\gamma}{(1 + \lambda) V^{*\infty}}\Bigr) x^{*} + \frac{1}{1 + \lambda} \Bigl(\frac{1}{V^{*\infty}} + \tau_{h,in}^{*}\Bigr)
    \end{aligned}
    \end{equation}

    \end{document} 

From -- https://tex.stackexchange.com/a/217524/197451-- is this your requirement

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\noindent Something\dots
\begin{alignat*}{2}
  K_{C} &= \sqrt{EG_{c}} &\qquad &\text{(plane stress)}\\
  \intertext{further description of the equation}
  K_{c} &= \sqrt{\frac{EG_{c}}{1-\nu^{2}}} &&\text{(plane strain)}
\end{alignat*}
Something\dots

\end{document}

OR

another solution with the help of manual raisebox

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\begin{equation}
\label{eq:distrib}
\begin{alignedat}{2}
n_c    &= \max[N \cdot z_{\max} \cdot x, 2 \cdot n_{\min}]
&\quad&\raisebox{-.5\normalbaselineskip}[0pt][0pt]{%
    where $x\sim U([0,1])$%
} \\
n_{co} &= \min[\max[n_c \cdot x, n_{\min}], n_c - n_{\min}] \\
n_{cp} &= n_c - n_{co} \\
n_f    &= N - n_c
\end{alignedat}
\end{equation}

\end{document}

from -- https://tex.stackexchange.com/a/391034/197451