How to align text vertically in table

Turning the comments of @leandris and myself into an answer.

The following uses \cmidrule and adjusts the \multirow accordingly. In addition it typesets the temperatures using the siunitx package (thanks to @Mico for pointing to the degree symbol used)

\documentclass[]{article}

\usepackage{booktabs}
\usepackage{multirow}
\usepackage{siunitx}

\begin{document}
\begin{table}[]
\centering
\begin{tabular}{lcccc}
\toprule
  \multicolumn{1}{c}{\multirow{2}{*}[-\dimexpr0.5\cmidrulewidth+\cmidrulesep]{Process}} & \multicolumn{4}{c}{Property at \SI{27}{\celsius}} \\ \cmidrule{2-5}
                                & S      & $\sigma$    & k       & zT       \\
\midrule
Unannealed                      & 140    & 1052        & 9.73    & 0.0636   \\
Annealed at \SI{1345}{\celsius} & 155    & 768         & 7.67    & 0.0722   \\
Annealed at \SI{1365}{\celsius} & 151    & 845         & 6.00    & 0.0964   \\
\bottomrule
\end{tabular}
\caption{Influence of annealing at \SI{27}{\celsius}}
\label{tab:influenceofannealing}
\end{table}
\end{document}

In addition to employing \cmidrule instead of \cline -- a suggestion already employed in @Skillmon's earlier answer -- I would also like to suggest that you employ the S column type (provided by the siunitx package) instead of the c column type for the four data columns. And, rather than using the typographically inappropriate º symbol for degrees (Celsius), do please write \SI{...}{\celsius}. Finally, I= would also like to suggest that you simplify the layout by placing the word "Process" into the top-left cell and make the header over the four data columns less wordy (and less-redundant, given that the information is already given in the caption).

\documentclass{article}
\usepackage{newtxtext,newtxmath} % Times Roman text and math font -- optional
\usepackage{booktabs,siunitx}
\newcolumntype{T}[1]{S[table-format=#1]}
\begin{document}
\begin{table}
\centering
\begin{tabular}{@{} l T{3.0}T{4.0}T{1.2}T{1.4} @{}}
\toprule
Process & \multicolumn{4}{c@{}}{Property} \\ 
\cmidrule(l){2-5}
& {S [\si{\micro\volt\per\kelvin}]}    
& {$\sigma$ [\si{\S\per\centi\meter}]} 
& {k [\si{\watt\per\meter\per\kelvin}]} 
& {zT}
\midrule
Unannealed                      & 140 & 1052 & 9.73 & 0.0636   \\
Annealed at \SI{1345}{\celsius} & 155 &  768 & 7.67 & 0.0722   \\
Annealed at \SI{1365}{\celsius} & 151 &  845 & 6.00 & 0.0964   \\
\bottomrule
\end{tabular}
\caption{Influence of annealing at \SI{27}{\celsius} (\SI{300}{\kelvin})}
\label{tab:influenceofannealing}
\end{table}
\end{document}

Here is yet another alternative using the cellspace package:

\documentclass{article}

\usepackage[column=O]{cellspace}
\usepackage{booktabs}
\usepackage{siunitx}
\usepackage{multirow}

\newlength\TopBottomLimit
\setlength\TopBottomLimit{0.7em}
\setlength\cellspacetoplimit{\TopBottomLimit}
\setlength\cellspacebottomlimit{\TopBottomLimit}

\begin{document}

\begin{table}
    \centering
    \begin{tabular}{O{l}O{c}O{c}O{c}O{c}}
        \toprule
        \multirow{2}{*}[-\TopBottomLimit]{Process} & \multicolumn{4}{O{c}}{Property at \SI{27}{\celsius}} \\ \cline{2-5}
                                        & \si{\celsius}      & $\sigma$    & k       & zT       \\
        \midrule
        Unannealed                      & 140    & 1052        & 9.73    & 0.0636   \\
        Annealed at \SI{1345}{\celsius} & 155    & 768         & 7.67    & 0.0722   \\
        Annealed at \SI{1365}{\celsius} & 151    & 845         & 6.00    & 0.0964   \\
        \bottomrule
    \end{tabular}
    \caption{Influence of annealing at \SI{27}{\celsius}}
    \label{tab:influenceofannealing}
\end{table}

\end{document}

Remark

You place the information that the process takes places at 27°C both in the header of the table and in the caption. I don't think that this redundancy contributes to a better readability of your document. You may think about reducing the degree of redundancy. In this case the issue will solve itself.