How to get the braces in the multirow spanning all three rows?

A simpler method is to use \bigdelim,

\documentclass{article}  
\usepackage{multirow,bigdelim}
\begin{document}
    \begin{table}[H]
        \centering
        \begin{tabular}{|l|c|r|}
            \hline
            \multicolumn{3}{|c|}{Number of Calls} \\
            \hline
            (6.128) & 6 & \rdelim\}{3}{20pt}[16]  \\                
            (2.016) & 3 &                         \\                
            (7.856) & 8 &                         \\                
            \hline
        \end{tabular}
    \end{table}
\end{document}

which gives

How about something like \multirow{3}{*}{$\left. \vphantom{\begin{tabular}{c}3\\3\\3\end{tabular}}\right\}$16}?

The scalerel package is also an alternative:

\documentclass{article}
\usepackage{scalerel}
\usepackage{multirow}
\begin{document}

\begin{table}[H]
\centering
\begin{tabular}{|l|r|r|l|c|r|r|}
\hline
Area & \multicolumn{2}{|c|}{Residents} & \multicolumn{3}{|c|}{Number of
Calls}     & Percentage Moving Calls \\
\hline
Zuidhorn & 6,878 & 38.3 & (6.128) & 6 &
\multirow{3}{*}{\scalerel*[1ex]{\}}{\rule[-2.8ex]{1ex}{7.2ex}}17} & 0\% \\
Aduard & 2,253 & 12.6 & (2.016) & 3 & & 0\%   \\
Lewenborg & 8,800 & 49.1 & (7.856) & 8 & & 0\%   \\
\cline{2-6}
Groningen & \multicolumn{2}{|c|}{171,201} & \multicolumn{3}{|c|}{300} &
40\%  \\ 
\cline{2-6}
Hoogkerk & 14,640 & 42.8\% & (12.84) & 13 &
\multirow{3}{*}{\scalerel*[1ex]{\}}{\rule[-2.8ex]{1ex}{7.2ex}}30} & 70\%
\\
Haren & 18,656 & 54.6\% & (16.38) & 16 & & 70\%  \\
Engelbert & 870 & 2.6 \% & (0.78) & 1 & & 70\%  \\
\hline \hline
Total & 223,298 & & & & &       \\
\hline
\end{tabular}
\caption{}
\label{}
\end{table}

\end{document}