In earlier releases of OpenELEC support for rsync was included. At some point it was moved to the unofficial addon repository. Unfortunately it looks like the unofficial addon repo has been abandoned. OpenElec 8 has been out for a year and there is still no unofficial addon support for it. I would assume that there won’t be any support for it in the future either.
I run OpenELEC on a RaspberryPi and really needed rsync support for it.
To get it working I cross compiled rsync for armv7 on my regular Linux machine and then copied the binary to my OpenElec box. I figured this might be handy for someone running into the same issue. This workflow is specific for ARM.