minipages have their own footnote counter, which is called mpfootnote.
When you use the option symbol* loading the package footmisc, it affects only the normal footnote counter.
So, if you want the same behavior for minipages, you have to add the line
\renewcommand\thempfootnote{\fnsymbol{mpfootnote}}
in your preamble.
MWE:
\documentclass{book}
\usepackage[symbol*,stable]{footmisc}
\usepackage[paperheight=5cm,showframe]{geometry}
\renewcommand\thempfootnote{\fnsymbol{mpfootnote}}
\begin{document}
\begin{minipage}{3in}
Here the text in environment\footnote{the footnote in minipage}
\end{minipage}
\vfil
\parbox{3in}{Here the text in parbox\footnotemark}
\footnotetext{the footnote in parbox}
\end{document}
Output:

EDIT
In response to your comment, if you want
- The
mpfootnote counter not to be reset at each minipage
- The
mpfootnote counter to share its value with the footnote counter
you can do the following:
Insert
\setcounter{mpfootnote}{\value{footnote}}
just after the beginning of each minipage, e.g.
\begin{minipage}{3in}
\setcounter{mpfootnote}{\value{footnote}}
Insert
\setcounter{footnote}{\value{mpfootnote}}
just before the end of each minipage, that is
\setcounter{footnote}{\value{mpfootnote}}
\end{minipage}
mpfootnotecounter "synchronize" withfootnotecounter? I mean that if thempfootnotecounter equal n, then thefootnotecounter followed equal n+1; vice versa. – user41022 Nov 17 '13 at 04:47