The llncs class uses a different way of defining theorem-like environments, so it's not a surprise that the other suggested methods don't work. Neither amsthm nor ntheorem based solution can work.
Here I assume the theorem to repeat precedes the repetition. I define a new environment reptheorem that stores the text; it has a mandatory argument that also sets a \label with the same name.
In the example I also show that the optional argument is honored.
\documentclass[envcountsect]{llncs}
\usepackage{environ}
\newcommand{\repeattheorem}[1]{%
\begingroup
\renewcommand{\thetheorem}{\ref{#1}}%
\expandafter\expandafter\expandafter\theorem
\csname reptheorem@#1\endcsname
\endtheorem
\endgroup
}
\NewEnviron{reptheorem}[1]{%
\global\expandafter\xdef\csname reptheorem@#1\endcsname{%
\unexpanded\expandafter{\BODY}%
}%
\expandafter\theorem\BODY\unskip\label{#1}\endtheorem
}
\begin{document}
\section{Title}
\begin{reptheorem}{foo}[Somebody]
Let $x$ be a foo. Then $x$ is also a baz.
\end{reptheorem}
\section{Another}
\repeattheorem{foo}
\end{document}

A feature asked in comments prompted to write “more modern code” for the job.
\documentclass[envcountsect]{llncs}
\usepackage{amsmath}
\newcommand{\replabel}{\label} % will be redefined in restatements
\ExplSyntaxOn
\NewDocumentCommand{\repeattheorem}{m}
{
\group_begin:
\renewcommand{\thetheorem}{\ref{#1}}
\renewcommand{\replabel}[1]{\tag{\ref{##1}}}
\prop_item:Nn \g_reptheorem_prop { #1 }
\endtheorem
\group_end:
}
\NewDocumentEnvironment{reptheorem}{m+b}
{
\prop_gput:Nnn \g_reptheorem_prop { #1 } { \theorem #2 \endtheorem }
\theorem#2\unskip\label{#1}\endtheorem
}{}
\prop_new:N \g_reptheorem_prop
\ExplSyntaxOff
\begin{document}
\section{Title}
\begin{reptheorem}{foo}[Somebody]
Let $x$ be a foo. Then $x$ is also a baz.
\begin{equation}\replabel{fooeq}
1=1
\end{equation}
with an equation.
\end{reptheorem}
\section{Another}
\repeattheorem{foo}
\end{document}
In theorems to be repeated, labels to equations (mandatory) should be set with \replabel. This way we can change the meaning to use \tag instead. The \label command cannot be used, because amsmath (necessary for \tag) would change back the meaning of \label in a display.
