Elbow grease is sweat? Indulge me in my unauthoritative protest. I always thought the point of "put some elbow grease into it" was that it referred to trying to remove a bolt that's stuck. If you haven't got any real grease to lubricate it with, all you can do is lean into it with more torque--elbow grease.
I understand it similarly: it's when you want to emphasize how a job requires human labors as opposed to just the oiled (greased) machinery doing all the works. The grease is symbolic, not anything on your elbow.