Changing footnote symbols

The footnote symbol code is actually very simple:

\def\@fnsymbol#1{\ensuremath{\ifcase#1\or *\or \dagger\or \ddagger\or
   \mathsection\or \mathparagraph\or \|\or **\or \dagger\dagger
   \or \ddagger\ddagger \else\@ctrerr\fi}}

So, if you were to copy that definition and replace \ddagger by \forall, then the third footnote would get an inverted A.

Do not forget to actually change the footnote counter to fnsymbol:

\renewcommand{\thefootnote}{\fnsymbol{footnote}}

As starting point the definition of \@fnsymbol can be used (latex.ltx, source2e.pdf: "21 Counter and Lengths". The following example goes a step further and removes the upper limit for the counter value. If needed the symbol will be multiplied (see the doubling of symbols in \@fnsymbol) with the help of package alphalph:

\documentclass{article}

\makeatletter
\newcommand*{\myfnsymbolsingle}[1]{%
  \ensuremath{%
    \ifcase#1% 0
    \or % 1
      *%   
    \or % 2
      \dagger
    \or % 3  
      \ddagger
    \or % 4   
      \mathsection
    \or % 5
      \mathparagraph
    \else % >= 6
      \@ctrerr  
    \fi
  }%   
}   
\makeatother

\newcommand*{\myfnsymbol}[1]{%
  \myfnsymbolsingle{\value{#1}}%
}

% remove upper boundary by multiplying the symbols if needed
\usepackage{alphalph}
\newalphalph{\myfnsymbolmult}[mult]{\myfnsymbolsingle}{}

\renewcommand*{\thefootnote}{%
  \myfnsymbolmult{\value{footnote}}%
}

\begin{document}
\footnote{a}\footnote{b}\footnote{c}\footnote{d}\footnote{e}%
\footnote{f}\footnote{g}\footnote{h}\footnote{i}\footnote{j}%
\footnote{k}\footnote{l}\footnote{m}\footnote{n}\footnote{o}%
\end{document}

You can use footmisc and define a set of symbols with \DefineFNsymbols (text-only symbols, i.e. no math) and \DefineFNsymbolsTM (for text or math symbols)

\DefineFNsymbolsTM{myfnsymbols}{% def. from footmisc.sty "bringhurst" symbols
  \textasteriskcentered *
  \textdagger    \dagger
  \textdaggerdbl \ddagger
  \textsection   \mathsection
  \textbardbl    \|%
  \textparagraph \mathparagraph
}%

And call it with

\setfnsymbol{myfnsymbols}

Here's a MWE

\documentclass{article}
\usepackage[symbol*]{footmisc}
\DefineFNsymbolsTM{myfnsymbols}{% def. from footmisc.sty "bringhurst" symbols
  \textasteriskcentered *
  \textdagger    \dagger
  \textdaggerdbl \ddagger
  \textsection   \mathsection
  \textbardbl    \|%
  \textparagraph \mathparagraph
}%
\setfnsymbol{myfnsymbols}
\begin{document}
\footnote{a}\footnote{b}\footnote{c}
\footnote{d}\footnote{e}\footnote{f}
\end{document}