\documentclass{article}

\usepackage{xstring}

\newcommand{\Client}[1]{\renewcommand{\Client}{#1}}

\makeatletter
\newcommand\foo[1]{%
  \ifnum\pdfstrcmp{#1}{Adam}=0 %
     \expandafter\@firstoftwo
  \else
    \expandafter\@secondoftwo
  \fi
  {Adam}
  {Eve}%
}
\makeatother

\begin{document}

\Client{Adam}

% Attempt using xstring
My client should be Adam: \IfSubStr{Adam}{\Client}{Adam}{Eve}

% Attempt using \pdfstrcmp
My client should be Adam: \foo{\Client}

\end{document}