Background image for minipage

You basically need to store the minipage content in a box register, measure it and overlay it with the graphic. The following environment awaits the image file name as first argument and then accepts any minipage arguments:

\documentclass{article}

\usepackage{graphicx}

\newsavebox\mysavebox
\newenvironment{imgminipage}[2][]{%
   \def\imgcmd{\includegraphics[width=\wd\mysavebox,height=\dimexpr\ht\mysavebox+\dp\mysavebox\relax,#1]{#2}}%
   \begin{lrbox}{\mysavebox}%
   \begin{minipage}%
}{%
   \end{minipage}
   \end{lrbox}%
   \sbox\mysavebox{\fbox{\usebox\mysavebox}}%
   \mbox{\rlap{\raisebox{-\dp\mysavebox}{\imgcmd}}\usebox\mysavebox}%
}

\begin{document}

\begin{imgminipage}{imagefilename}{5cm}
    Some text\\
    Hello world!
\end{imgminipage}

\end{document}

---

I added now a bgimage key to adjustbox (develop version) which allows you to add a background image. To have also a \fbox use either:

\begin{adjustbox}{minipage=<width>,fbox,bgimage=<image>}
    <text>
\end{adjustbox}

which adds the \fbox first and then places the background image behind it, so that the frame covers some small part of the image, or

\begin{adjustbox}{minipage=<width>,margin=\fboxsep,bgimage=<image>,frame}
    <text>
\end{adjustbox}

which adds the same margin manually, places the background image and then draws a tight frame around it, so that the image is fully shown (and a little bit smaller than before).

\documentclass{article} 
\usepackage[english]{babel}
\usepackage{graphicx}
\usepackage{blindtext}
\newsavebox\MBox
\newenvironment{Minipage}[1]
  {\par\smallskip\begin{lrbox}{\MBox}\begin{minipage}{#1}}
  {\end{minipage}\end{lrbox}%
   \makebox(0,0){\put(0,0){%
     \includegraphics[width=\wd\MBox,height=2\ht\MBox]{tiger}}}%
   \usebox\MBox\par%
  }
\begin{document}
Some text before

\begin{Minipage}{0.5\textwidth} 
\blindtext
\end{Minipage}

Some text behind    
\end{document}

EDIT:

Here a modification of the package mdframed. So you can use all the settings which are provided by mdframed according to the background image:

\documentclass{article} 
\usepackage[english]{babel}
\usepackage{graphicx,tikz}
\usepackage{blindtext}
\usepackage[backgroundcolor=yellow!10,style=0]{mdframed}
\makeatletter
\newrobustcmd*\mdf@backgroundimage{%
      \rlap{\hspace*{0.5\mdfboundingboxwidth}%
             \makebox[0pt][c]{%
               \tikz[remember picture]%
                  \node (0,0) [opacity=0.4] {%
                  \includegraphics[width=\mdfboundingboxwidth,%
                                height=\mdfboundingboxheight,%
                                keepaspectratio]%
                                {\backgroundimage}%
                              };
             }%
           }%
}
\newenvironment{Minipage}[2][]
  {\def\backgroundimage{#2}%
   \appto\md@frame@background@single\mdf@backgroundimage%
   \appto\md@frame@background@first\mdf@backgroundimage%
   \appto\md@frame@background@middle\mdf@backgroundimage%
   \appto\md@frame@background@second\mdf@backgroundimage%
   \begin{mdframed}[#1]%
   }
  {\end{mdframed}}
\makeatother
\begin{document}
Some text before

\begin{Minipage}{tiger} 
\blindtext
\end{Minipage}

Some text behind    

\begin{Minipage}{tiger} 
\blindtext[10]
\end{Minipage}
\end{document}

EDIT 2 Thanks to xport -- I added the option opacity by using tikz.

A possibility is to combine minipage with includegraphics. I tried it with the following configuration (option demo and color{red}. So I can compile without any image :-)

\documentclass[demo]{article}
\usepackage{adjustbox}
\usepackage{xcolor}
\usepackage{graphicx}
\usepackage{lipsum}
\newlength{\widthofminipage}
\newsavebox{\MyMinBackBox}
% minipageback{<width>}{<backgroundimage>}
\newenvironment{minipageback}[2]{%
      \setlength{\widthofminipage}{#1}%
      \def\pictureminback{#2}%
      \begin{lrbox}{\MyMinBackBox}%
      \begin{minipage}[b]{\widthofminipage}\color{red}%
    }{%
     \end{minipage}\end{lrbox}%
     \includegraphics[width=\wd\MyMinBackBox,height=\dimexpr\ht\MyMinBackBox+\dp\MyMinBackBox\relax]{\pictureminback}%
    \llap{\usebox{\MyMinBackBox}}%
}

\begin{document}
Text 


\begin{minipageback}{.5\textwidth}{logo}
\lipsum[1]
\end{minipageback}
\end{document}