I finally found a way to insert a footnote inside a caption while using float:Table in LyX, from this answer. You need to insert \protect\footnotemark in the caption and \footnotetext{Text to footnote} directly after the float.
See source:
\begin{table}[h]
\begin{tabular}{|c|c|}
\hline
1 & 2\tabularnewline
\hline
\hline
3 & 4\tabularnewline
\hline
\end{tabular}
\caption{This is the caption. \protect\footnotemark}
\end{table}
\footnotetext{Text to footnote}
Is there a more efficient way to include a footnote inside a caption when using LyX?
\footnotemarkwithin a float and matching\footnotetextoutside the float could end up on different pages. rather counterproductive. – barbara beeton Dec 12 '12 at 21:24minipage(Insert --> Box --> Frameless) as in http://tex.stackexchange.com/a/84331/586 – Torbjørn T. Jul 31 '13 at 20:14