Discussion Forum—A Way with Words, a fun radio show and podcast about language

In 1836, the Western Monthly Magazine and Literary Journal printed "This was unlucky, for as I had to do all my caulking and carpentering myself, I knew I could not get her done by afternoon, and I had made an engagement with Ned Harris and another smart chap for a sail." 

There are also numerous cites in 1865 congressional hearings containing get her done, and they, too, are referring to finishing boats or ships.

I believe the expression comes from calling ships her and morphed into a meaning of her as this here job.

Interesting ngram here, if you care to see it and explore.