thmtools
provides an environment called restatable
with the following syntax
\begin{restatable}[theorm-title]{envname}{commandtorestate}
<some text>
\end{restatable}
On page 7 of the documentation we have to following example
\documentclass{article}
\usepackage{thmtools, thm-restate}
\declaretheorem{theorem}
\begin{document}
\begin{restatable}[Euclid]{theorem}{firsteuclid}
\label{thm:euclid}%
For every prime $p$, there is a prime $p'>p$.
In particular, the list of primes,
\begin{equation}\label{eq:1}
2,3,45,7,\dots
\end{equation}
is infinite.
\end{restatable}
\firsteuclid*
\vdots
\firsteuclid*
\end{document}
which produce the following

If I understand you correctly, the following patch is what you want
\documentclass{article}
\usepackage{thmtools,thm-restate}
\declaretheorem{theorem}
\usepackage{regexpatch}
\makeatletter
\xpatchcmd\thmt@restatable{%
\csname #2\@xa\endcsname\ifx\@nx#1\@nx\else[{#1}]\fi
}{%
\ifthmt@thisistheone
\csname #2\@xa\endcsname\ifx\@nx#1\@nx\else[{#1}]\fi
\else
\csname #2\@xa\endcsname[{Restated}]
\fi}{}{}
\makeatother
\begin{document}
\begin{restatable}[Euclid]{theorem}{firsteuclid}
\label{thm:euclid}%
For every prime $p$, there is a prime $p'>p$.
In particular, the list of primes,
\begin{equation}\label{eq:1}
2,3,45,7,\dots
\end{equation}
is infinite.
\end{restatable}
\firsteuclid*
\vdots
\firsteuclid*
\end{document}
