My baby names book (which is semi-reliable!) lists Marilla as an English elaboration of
Mary. Seems about right to me.
http://www.justbabynames.com/ reckons it's a variation of
Muriel, Celtic, meaning "Shining Sea". A few different sites share this definition.
http://www.20000-names.com/ thinks it's a flower name - they don't say it but maybe from
Amaryllis?
I'm betting on the
Mary origin. :-)