Insert image and list inside a table

You can use \raisebox to adjust the vertical positioning of the image:

\documentclass{memoir}
\usepackage[demo]{graphicx}% delete the demo option in your actual code
\usepackage{enumitem}
\usepackage{booktabs}

\begin{document}

\begin{table}[h!]
     \begin{center}
     \begin{tabular}{ c  p{5cm}  p{5cm}  }
     \toprule
      my.Lboro & Advantages & Disadvantages \\ 
    \cmidrule(r){1-1}\cmidrule(lr){2-2}\cmidrule(l){3-3}
     \raisebox{-\totalheight}{\includegraphics[width=0.3\textwidth, height=60mm]{images/myLboro.png}}
      & 
      \begin{itemize}[topsep=0pt]
      \item Accessibility
      \item Up to date information
      \item Fulfil students needs and wants \ldots
      \end{itemize}
      & 
      \begin{itemize}[topsep=0pt]
      \item Accessibility
      \item Up to date information
      \item Fulfil students needs and wants \ldots
      \end{itemize}
      \\ \bottomrule
      \end{tabular}
      \caption{my.Lboro Analysis}
      \label{tbl:myLboro}
      \end{center}
      \end{table}

\end{document}

I also made some changes to your code (as suggestions, of course):

1. I used the booktabs package to improve the table design (in particular, no vertical rules).
2. I used the enumitem package to suppress some vertical spacing before the lists.

The array package provides the m{<len>} column type which vertically aligns content in rows:

\documentclass{article}
\usepackage{graphicx}% http://ctan.org/pkg/graphicx
\usepackage{array}% http://ctan.org/pkg/array
\begin{document}

\begin{table}[h!]
  \centering
  \begin{tabular}{ | c | m{5cm} | m{5cm} | }
    \hline
    my.Lboro & Advantages & Disadvantages \\ \hline
    \begin{minipage}{.3\textwidth}
      \includegraphics[width=\linewidth, height=60mm]{tiger}
    \end{minipage}
    &
    %\begin{minipage}[t]{5cm}
      \begin{itemize}
        \item Accessibility
        \item Up to date information
        \item Fulfil students needs and wants \ldots
      \end{itemize}
    %\end{minipage}
    & 
    %\begin{minipage}{5cm}
      \begin{itemize}
        \item Accessibility
        \item Up to date information
        \item Fulfil students needs and wants \ldots
      \end{itemize}
    %\end{minipage}
    \\ \hline
  \end{tabular}
  \caption{my.Lboro Analysis}\label{tbl:myLboro}
\end{table}

\end{document}
​

Then, additionally, you can place the image inside a minipage of equivalent size (box it) to obtain the correct alignment. The adjustbox package provides a similar alignment modification fox boxes.