Table captions continued

The package longtable allows you to define

1. a header for the first page by \endfirsthead
2. a header for all next pages by \endhead
3. a footer fo all pages expect the last one by \endfoot
4. a footer for the last page by \endlastfoot.

In your case you can do:

\documentclass{article}

\usepackage[paper=a6paper]{geometry}
\usepackage{booktabs}
\usepackage{longtable}
\usepackage[font=small,labelfont=bf]{caption}

\begin{document}

\begin{longtable}{ccc}
\caption {Some Members of the Suggested Estimator}\label{Tab:1}\\\toprule
\endfirsthead
\caption* {\textbf{Table \ref{Tab:1} Continued:} Blablabla}\\\toprule
\endhead
\endfoot
\bottomrule
\endlastfoot
A subset of $\overline{y}_{NH}$ & $\textbf{a}$ & $\textbf{b}$\\
\midrule
test1 & 1 & 0 \\
\midrule
test2 & 1 & 1 \\
\midrule
test3 & \newline 1 & \newline $\beta_{2} \left(x\right)$ \\
\midrule
A subset of $\overline{y}_{NH}$ & $\textbf{a}$ & $\textbf{b}$\\
\midrule
test4 & $C_{x} $ & $\rho_{yx} $ \\
\midrule
test5 & $\rho _{yx} $ & $C_{x} $ \\
\midrule
test6 & $\rho _{yx} $ \\
A subset of $\overline{y}_{NH}$ & $\textbf{a}$ & $\textbf{b}$\\
\midrule
test1 & 1 & 0 \\
\midrule
test2 & 1 & 1 \\
\midrule
test3 & \newline 1 & \newline $\beta_{2} \left(x\right)$ \\
\midrule
A subset of $\overline{y}_{NH}$ & $\textbf{a}$ & $\textbf{b}$\\
\midrule
test4 & $C_{x} $ & $\rho_{yx} $ \\
\midrule
test5 & $\rho _{yx} $ & $C_{x} $ \\
\midrule
test6 & $\rho _{yx} $ \\
\end{longtable}

\end{document}