According to the reledmac manual, optional arguments can be provided to either \& for inserting stuff after a verse or to \stanza for inserting stuff before a verse. Thus, either of these could be used for creating a blank line by adding a mere space.
\documentclass{article}
\usepackage{reledmac}
\setstanzaindents{1,0}
\setcounter{stanzaindentsrepetition}{1}
% doesn't word anymore
%\def\endstanzaextra{\pstart x\strut\skipnumbering\pend}
\begin{document}
\beginnumbering
\stanza First Verse & Second Verse \&[*After verse*]
\stanza[/Before verse/] Third Verse & Fourth Verse \&
\stanza[ ]Third Verse & Fourth Verse \&[ ]
\stanza Third Verse & Fourth Verse \&
\endnumbering
\end{document}

Looking at the style file for ways to automate things, a blank line can automatically be added at the beginning of each stanza with the redefinition
\makeatletter
\renewcommandx{\stanza}[1][1,usedefault]{\@startstanza[#1~]}
\makeatother
Likewise, a blank line can be automatically added at the end of each stanza with this redefinition:
\makeatletter
\renewcommandx{\@stopstanza}[1][1,usedefault]{%
\unskip%
\endlock%
\pend[#1$\!\!$]%
\endgroup%
\instanzafalse%
}
\makeatother
Both of these redefinitions merely modify the default optional argument of either \stanza or \&.
In use:
\documentclass{article}
\usepackage{reledmac}
\setstanzaindents{1,0}
\setcounter{stanzaindentsrepetition}{1}
\makeatletter
% THE FOLLOWING INSERTS BLANK LINE BEFORE EACH STANZA
%\renewcommandx{\stanza}[1][1,usedefault]{\@startstanza[#1~]}
% THE FOLLOWING INSERTS BLANK LINE AFTER EACH STANZA
\renewcommandx{\@stopstanza}[1][1,usedefault]{%
\unskip%
\endlock%
\pend[#1$\!\!$]%
\endgroup%
\instanzafalse%
}
\makeatother
\begin{document}
\beginnumbering
\stanza[\textsf{optional before}] First Verse & Second Verse \&[\textsf{optional after}]
\stanza Third Verse & Fourth Verse \&
\stanza Fifth Verse & Sixth Verse \&
\stanza 7th Verse & 8th Verse \&
\endnumbering
\end{document}

\AtEveryPendinstead. In general, reledmac/reledpar is conceveided to avoid overriding. – Maïeul Apr 06 '16 at 14:47\AtEveryPend{}seems to perform its action on every verse, not just every stanza. – Steven B. Segletes Apr 06 '16 at 14:51\newcommand{\mystanza}{\stanza[mycontent]}. – Maïeul Apr 06 '16 at 15:37\patchcmd. – Maïeul Apr 06 '16 at 15:38