Horizontally centering certain columns in a table to the page

Edit: After OP comment below can it seems that he wish that the column with "material" in column head be placed in the middle of text width. This can be achieved by shifting table to the left side. This can be done on many ways ... with use of invisible rule after threeparttable cross my mind as the simplest. Accordingly I corrected my MWE from the first version of answer:

\documentclass{article}
\usepackage{booktabs, multirow, threeparttable}
\usepackage{siunitx}
\newlength\tabshift% added

%-------------------------------------------- to show page layout
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}
\begin{table}[htpb] 
    %\footnotesize
    \centering
    \begin{threeparttable}
        \caption{Material properties of cladding alloys compared to structural alloys at \SI{705}{\celsius}}
        \label{tab:properties}
        \begin{tabular}{rlS}
             \cmidrule[0.08em]{2-3}
            & Material & {CTE (\SI{e-3}{\per\celsius})} \\
            \cmidrule[0.05em]{2-3}
            \multirow{2}{*}{Structural alloys \bigg\{} & 316H & 18.8 \\
            & 800H & 17.2 \\
            \cmidrule[0.05em]{2-3}
            \multirow{3}{*}{Claddings \Bigg\{} & Ni-201 & 15.8  \\
            & Hastelloy N & 14.3  \\
            & Haynes 242 & 13  \\
            \cmidrule[0.08em]{2-3}
        \end{tabular}
    \end{threeparttable}%
        \rule{0.5\tabshift}{0pt}% added
\end{table}
\end{document}

The last table column I change to S column type using package siunitx. You can make detailed specification for S column, for example as S[table-format=2.1] (recommended to do). In your particular case the result is the same in the both cases.

Note (1): Please, in future always provide small but complete document (MWE: Minimal (non)Working Example) as I do above. Help people who are willing to help you. it is not fun to write missing preamble ...

Note (2): Please, in future be more accurate in formulating your question :). You receive two answers both based -- regarding your comment below -- on wrong assumption. And by the way, original position of table (before shifting to the left) seems to bi just fine ...

Fine tuning of table positioning you can do with change of factor 0.5 in \rule{0.5\tabshift}{0pt} to value which to your taste place table the best on the page.

Addendum: Form better visual centering of table on page I would redesign your table as follows:

\documentclass{article}
\usepackage{booktabs, makecell, multirow, threeparttable}
\usepackage{siunitx}

%---------------------------------------------------------------%
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}
\begin{table}[htpb]
    \centering
    \begin{threeparttable}
        \caption{Material properties of cladding alloys compared to structural alloys at \SI{705}{\celsius}}
        \label{tab:properties}
        \begin{tabular}{rlS}
             \cmidrule[0.08em]{2-3}
            & Material & {CTE (\SI{e-3}{\per\celsius})} \\
            \cmidrule[0.05em]{2-3}
            \multirow{2}{*}{\makecell{Structural\\ alloys} \bigg\{} & 316H & 18.8 \\
            & 800H & 17.2 \\
            \cmidrule[0.05em]{2-3}
            \multirow{3}{*}{Claddings \Bigg\{} & Ni-201 & 15.8  \\
            & Hastelloy N & 14.3  \\
            & Haynes 242 & 13  \\
            \cmidrule[0.08em]{2-3}
        \end{tabular}
    \end{threeparttable}
\end{table}
\end{document}

The table will look unbalanced anyway.

You can do it without \multirow; by accepted convention, blank cells mean repeated value.

\documentclass{article}
\usepackage{booktabs, multirow, threeparttable}
\usepackage{siunitx}

\begin{document}

\begin{table}[htpb]
\centering
\begin{threeparttable}

\caption{Material properties of cladding alloys compared 
  to structural alloys at \SI{705}{\celsius}}
\label{tab:properties}

\begin{tabular}{@{}llS[table-format=2.1]@{}}
\toprule
Type & Material & {CTE (\SI{e-3}{\per\celsius})} \\
\midrule
Structural alloy & 316H        & 18.8 \\
                 & 800H        & 17.2 \\
\midrule
Cladding         & Ni-201      & 15.8 \\
                 & Hastelloy N & 14.3 \\
                 & Haynes 242  & 13   \\
\bottomrule
\end{tabular}

\end{threeparttable}

\end{table}

\end{document}

The alternative doesn't look as appealing:

\documentclass{article}
\usepackage{booktabs, multirow, threeparttable}
\usepackage{siunitx}

\begin{document}
\begin{table}[htpb]
    \centering
    \begin{threeparttable}
        \caption{Material properties of cladding alloys compared to structural alloys at \SI{705}{\celsius}}
        \label{tab:properties}
        \begin{tabular}{@{}r@{}lc}
             \cmidrule[0.08em]{2-3}
            & Material & CTE (\SI{e-3}{\per\celsius}) \\ 
            \cmidrule[0.05em]{2-3}
            \multirow{2}{*}{\llap{Structural alloys \bigg\{\ }} & 316H & 18.8 \\
            & 800H & 17.2 \\ 
            \cmidrule[0.05em]{2-3}
            \multirow{3}{*}{\llap{Claddings \Bigg\{\ }} & Ni-201 & 15.8  \\ 
            & Hastelloy N & 14.3  \\ 
            & Haynes 242 & 13  \\ 
            \cmidrule[0.08em]{2-3}
        \end{tabular}
    \end{threeparttable}
\end{table}
\end{document}