How to save and restore the nodes between tikzpictures?

If I understand correctly, you want to use the relative positions of the saved nodes in your new picture. That is, each node should refer to a position in the new picture relative to the new origin.

Here's some code that saves all the data for a list of specified nodes which can then be restored at a later time in the document. It uses LaTeX3 stuff internally since it's more about programming and expansion and L3 makes that sooo much easier.

\documentclass{article}
%\url{http://tex.stackexchange.com/q/307356/86}
\usepackage{tikz} 
\usepackage{xparse}

\ExplSyntaxOn

% We save our information in a ``property list'', which is L3's
% version of an associative array or dictionary.  They keys will give
% the ability to store several groups of nodes and restore them at
% will.
\prop_new:N \g__sn_prop
% We'll need a token list for constructing the saved data.
\tl_new:N \l__sn_tmpa_tl

% This is the command that actually does the work.  It constructs a
% token list which contains the code that will restore the node data
% when invoked.  The two arguments are the name of this group (for
% reference later) and a comma separated list of the node names to be
% saved.
\cs_new_nopar:Npn \save_nodes:nn #1#2
{
  % Clear our token list
  \tl_clear:N \l__sn_tmpa_tl
  % Iterate over the list of node names
  \clist_map_inline:nn {#2}
  {
    % Before we start trying to save the node, check that it exists.
    % The macro \pgf@sh@ns@nodename is only defined if that node exists.
    \tl_if_exist:cT {pgf@sh@ns@##1}
    {
      % The node information is stored in a series of macros of the form
      % \pgf@sh@XX@nodename where XX is one of the following.
      \clist_map_inline:nn {ns,np,ma,nt,pi}
      {
        % Our token list will look like:
        %
        % \tl_set:cn {pgf@sh@XX@nodename} {<current contents of that macro>}
        %
        % This will restore \pgf@sh@XX@nodename to its current value
        % when this list is invoked.
        %
        % This part puts the \tl_set:cn {pgf@sh@XX@nodename} in place
        \tl_put_right:Nn \l__sn_tmpa_tl
        {
          \tl_set:cn {pgf@sh@####1@##1}
        }
        % Now we put the current contents in place.  We're doing this in
        % an expansive context to get at the contents.  The \exp_not:v
        % part takes the current value of \pgf@sh@XX@nodename and puts
        % it in place, preventing further expansion.
        \tl_put_right:Nx \l__sn_tmpa_tl {{\exp_not:v {pgf@sh@####1@##1}}}
      }
    }
  }
  % Once we've assembled our token list, we store it in the property
  % list using the key we were given.
  \prop_gput:NnV \g__sn_prop {#1} \l__sn_tmpa_tl
}

\cs_new_nopar:Npn \restore_nodes:n #1
{
  % Restoring nodes is simple: look in the property list for the key
  % and if it exists, invoke the macro stored there.
  \prop_get:NnNT \g__sn_prop {#1} \l__sn_tmpa_tl
  {
    \tl_use:N \l__sn_tmpa_tl
  }
}

% These two are wrappers around our internal commands.
%
% The first argument is the label for our group of nodes (so that we
% can refer to them later) and the second argument is a comma
% separated list of nodes to save.
\DeclareDocumentCommand \SaveNodes {m m}
{
  \save_nodes:nn {#1}{#2}
}

% The argument to this is the label for our group of nodes to restore.
\DeclareDocumentCommand \RestoreNodes {m}
{
  \restore_nodes:n {#1}
}

\ExplSyntaxOff

\begin{document}

\begin{tikzpicture}
\coordinate (a) at (1,2);
\coordinate (b) at (1,3);
\coordinate (c) at (1,4);
% save
\SaveNodes{here}{a,b,c}
\end{tikzpicture}

\begin{tikzpicture}
\fill[black] (0,0) circle (1.5 pt);
  \foreach \c in {a,...,c}{ \fill[red] (\c) circle (1.5 pt); }
\end{tikzpicture}

\begin{tikzpicture}
\coordinate (a) at (2,2);
\coordinate (b) at (2,3);
\coordinate (c) at (2,4);
\fill[black] (0,0) circle (1.5 pt);
 \foreach \c in {a,...,c}{ \fill[blue] (\c) circle (1.5 pt); }
\end{tikzpicture}

%restore I would like to use the same nodes like at the beginning
\begin{tikzpicture}
\RestoreNodes{here}
\fill[black] (0,0) circle (1.5 pt);
  \foreach \c in {a,...,c}{ \fill[red] (\c) circle (1.5 pt); } % with 
\end{tikzpicture}
\end{document}

I added the black circles at the origin to show that it is the original nodes that are used, not the new ones. I think I'm saving all the information about the nodes, not just their coordinates, so this solution would work with general nodes not just coordinates. With a little more work, it would be possible to take a list of nodes at the restore stage and only restore those.

Here's a non-L3 version. It works on a per-node basis (but you could probably wrap it if the nodes always have the same names).

\makeatletter

\def\@savecoord#1#2{%
  \expandafter\let\expandafter\sn@temp\csname pgf@sh@#1@#2\endcsname
  \expandafter\def\expandafter\sn@tempb\expandafter{%
    \expandafter\gdef\csname pgf@sh@#1@#2\expandafter\endcsname\expandafter{%
      \sn@temp
    }%
  }%
  \expandafter\g@addto@macro\expandafter\sn@tempa\expandafter{\sn@tempb}%
}

% {ns,np,ma,nt,pi}
\def\savecoordinate#1#2{%
  \def\sn@tempa{}%
  %
  \@savecoord{ns}{#2}%
  \@savecoord{np}{#2}%
  \@savecoord{ma}{#2}%
  \@savecoord{nt}{#2}%
  \@savecoord{pi}{#2}%
%
  \expandafter\global\expandafter\let\csname sn@#1\endcsname\sn@tempa
}

\def\restorecoordinate#1{%
  \csname sn@#1\endcsname
}

\makeatother