There are many ways to do it. The following also works with plain TeX and without e-TeX or other extensions. However, it is not expandable, but this does not seems to be needed here.
\documentclass{article}
\newcommand*{\dia}[1]{%
\begingroup % keep macro definitions local
\def\param{#1}%
\def\monday{monday}%
\ifx\param\monday
Tuesday%
\else
Rest%
\fi
\endgroup
}
\begin{document}
\dia{monday}
\end{document}

Links:
An expandable solution without direct string comparison:
\documentclass{article}
\makeatletter
\@namedef{dia@Monday}{Tuesday}
\@namedef{dia@Tuesday}{Wednesday}
\@namedef{dia@Wednesday}{Thursday}
\@namedef{dia@Thursday}{Friday}
\@namedef{dia@Friday}{Saturday}
\@namedef{dia@Saturday}{Sunday}
\@namedef{dia@Sunday}{Monday}
\newcommand*{\dia}[1]{%
\expandafter\ifx\csname dia@#1\endcsname\relax
Rest%
\else
\csname dia@#1\endcsname
\fi
}
\makeatother
\begin{document}
\dia{Monday}
\dia{\dia{Monday}}
\dia{\dia{\dia{Monday}}}
\dia{\dia{\dia{\dia{Monday}}}}
\dia{\dia{\dia{\dia{\dia{Monday}}}}}
\dia{\dia{\dia{\dia{\dia{\dia{Monday}}}}}}
\dia{\dia{\dia{\dia{\dia{\dia{\dia{Monday}}}}}}}
\dia{\dia{\dia{\dia{\dia{\dia{\dia{\dia{Monday}}}}}}}}
\end{document}

etoolbox:\usepackage{etoolbox} \newcommand*{\dia}[1]{\ifstrequal{#1}{monday}{tuesday}{rest}}– moewe May 04 '18 at 11:33