How to center the minipage?

Put it inside a centered box

\documentclass{article}
\begin{document}
\noindent\hrulefill\par
\noindent\makebox[\textwidth][c]{%
    \begin{minipage}{1.5\textwidth}
        \hrulefill\par
    \end{minipage}}
\end{document}

A simple way to do this is to use \centerline{..} around the minipage. (This does not allow for verbatim or similar special content, but there is a \Centerline variant from the realboxes package, which does!)

I would add some small or medium vertical skip before and after this minipage to get decent vertical spacing.

\documentclass{article}
\usepackage{lipsum}% dummy text
\begin{document}

\lipsum[1]

\par\smallskip\noindent
\centerline{\begin{minipage}{1.5\textwidth}
\lipsum[2]
\end{minipage}}
\par\smallskip

\lipsum[3]

\end{document}

Another easy way is to use the adjustbox environment from the package with the same name. It provides the keys minipage=<minipage args>, center(=<length>, normally \linewidth) and also margin(=<left/right> <top/bottom>, and more).

\documentclass{article}
\usepackage{adjustbox}
\usepackage{lipsum}% dummy text
\begin{document}

\lipsum[1]

\begin{adjustbox}{minipage=1.5\textwidth,margin=0pt \smallskipamount,center}
\lipsum[2]
\end{adjustbox}

\lipsum[3]

\end{document}

Compare the similar question and answers at How can I center a too wide table?.

I was putting parcolumns and verbatim blocks inside my minipage which was causing problems. I found this rather simple solution quite effective:

\noindent\hspace{0.15\linewidth}\begin{minipage}{0.7\linewidth}
...
\end{minipage}

This creates a smaller minipage and centres with a positive hspace. Using the rule hspace = (1\linewidth - minipage_width) / 2 will also work for values larger than \linewidth and produce a negative hspace:

\newlength{\myminipagewidth}
\newlength{\myminipagecentering}
\setlength{\myminipagewidth}{1.5\linewidth} %change this
\setlength{\myminipagecentering}{(\linewidth-\myminipagewidth)/2}
\noindent\hspace{\myminipagecentering}\begin{minipage}{\myminipagewidth}
...
\end{minipage}

(sub \linewidth with \textwidth when needed)