Inserting a small vertical space in a table

use

\rule{0pt}{4ex}    

in the first column of that line.

For a general reference on how to improve the spacing in tabular and array lines, see the article "Correct spacing for tables and arrays" by Claudio Beccari on p. 10 of TeX and TUG News 1993 (Vol. 2, No. 3).

His method, which involves judiciously inserting "struts", applies to lines in tabular (as well as tabular*, supertabular, xtabular, longtable) and array environments which contain

• superscript material, on a line that's preceded by an \hline,
• subscript material, on a line that's followed by an \hline, and
• any other lines with material (including \hlines) above or below them that might result in a cramped look of the output.

He suggested defining a "top strut" and a "bottom strut" as follows:

\newcommand\T{\rule{0pt}{2.6ex}}       % Top strut
\newcommand\B{\rule[-1.2ex]{0pt}{0pt}} % Bottom strut

Using your MWE as a starting point -- by the way, 9pt is not a recognized option in the article document class, so I'm omitting it -- one could put these macros to use as follows:

\documentclass[letterpaper]{article}
\usepackage{helvet}
\renewcommand{\familydefault}{\sfdefault}

\newcommand\T{\rule{0pt}{2.6ex}}       % Top strut
\newcommand\B{\rule[-1.2ex]{0pt}{0pt}} % Bottom strut

\begin{document}  
\begin{tabular}{ p{8em}  r  r  l }  
\hline      
          & Total     & Average   & Unit \T\B  \\    \hline      
Area 1    & 419773    &   9.15    &  m\textsuperscript{2} \T \\      
Volume 1  & 0         &      0    &  m\textsuperscript{3} \B \\    \hline  
\end{tabular}

\end{document}

The simplest solution was given by David Carlisle in How to add vertical space struts after hline?

\hline
\noalign{\vskip 2mm}    

For those who can use the bigstrut package, then just inserting \bigstrut[t] will fix the problem.

\documentclass[letterpaper]{article}
\usepackage{helvet}
\usepackage{bigstrut}
\renewcommand{\familydefault}{\sfdefault}

\begin{document}
\begin{picture}(0,0)

\put(20,-136){\mbox{    
        \footnotesize   
        \begin{tabular}{ p{8em}  r  r  l }  
        \hline      
                        & Total     & Average   & Unit   \\
        \hline      
        Area1           & 419773    &   9.15        &  \emph{m$^2$} \bigstrut[t]     \\  
        Area2           & 0     &   0       &  \emph{m$^3$}      \\
        \hline  
        \end{tabular}
    }}  

\end{picture}

\end{document}