Without using the LaTeX preamble to insert some packages like threeparttable or ctable and without edit the whole table in an ERT box in order to use alternatives to the standard tabular environment, a simple solution is insert in the table float a minipage as wide as the table must be, then make a normal table with normal footnotes. The problem is that the normal footnote rule have no sense at the bottom of a table, but this can be solved with a ERT box with \renewcommand\footnoterule{...}.
Screenshot of LyX and the resulting PDF are self explanatory. (To see at original size open the image in another browser window. Spanish version, sorry):

Complete LaTeX source generated by Lyx:
% Vista preliminar del código fuente
%% LyX 2.1.2 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[english]{article}
\usepackage[T1]{fontenc}
\usepackage[latin9]{inputenc}
\usepackage{array}
\makeatletter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
%% Because html converters don't know tabularnewline
\providecommand{\tabularnewline}{\\}
\makeatother
\usepackage{babel}
\begin{document}
\begin{table}
\protect\caption{A table}
\renewcommand\footnoterule{ \kern -1ex}
\begin{minipage}[t]{1\columnwidth}%
\begin{center}
\begin{tabular*}{1\linewidth}{@{\extracolsep{\fill}}>{\centering}p{0.3\linewidth}c>{\centering}p{0.3\linewidth}}
\hline
Aaa & Bbb & Cccc\tabularnewline
\hline
11%
\footnote{A footnote about 11%
} & 12 & 13\tabularnewline
21 & 22%
\footnote{Another footnote%
} & 23\tabularnewline
\hline
\end{tabular*}
\par\end{center}%
\end{minipage}
\end{table}
\end{document}
If you want table footnotes as normal footnotes with continuous numeration, one option could be the footnote package (insret \usepackage{footnote} in Document > Configuration LaTeX preamble and the savenotes environment (use ERT boxes to insert \begin{savenotes} and \end{savenotes}).
MWE in LaTeX:

\documentclass[a5paper,w]{article}
\usepackage{footnote}
\begin{document}
Some text\footnote{A text footnote}
\begin{savenotes}
\begin{table}[!h]
\centering
\caption{A table}
\renewcommand\footnoterule{ \kern -1ex}
\begin{tabular}{lll}
\hline Aaa & Bbb & Cccc\\
\hline 11 \footnote{A table note about 11 cell} & 12 & 13\\
21 & 22 \footnote{A table note about 22} & 23\\
\hline
\end{tabular}
\end{table}
\end{savenotes}
\begin{savenotes}
\begin{table}[!h]
\centering
\caption{A table}
\renewcommand\footnoterule{ \kern -1ex}
\begin{tabular}{lll}
\hline Aaa & Bbb & Cccc\\
\hline 31 \footnote{A table note about 31} & 32 & 33\\
41 & 42 \footnote{One more table note about 42} & 43\\
\hline
\end{tabular}
\end{table}
\end{savenotes}
Some more text\footnote{Another text footnote}
\end{document}
threeparttableorctablepackages. – Gonzalo Medina Jun 25 '15 at 15:43