Grammars: is there some connection between non-terminals $S$ and $S'$?
$S'$ is just another nonterminal.

The reason $S'$ is used instead of, say, $T$, is that $S$ has been established as standard  for the start symbol of a grammar. Now, if you are talking about an algorithm that transforms a grammar (e.g. into a normal form), you will need to distinguish between the start symbol of the original grammar and that of the transformed grammar. In order to indicate that both are start symbols, it has become the norm to use $S'$ for one of them (usually the latter).