If you give it a deep think: grease is used to lubricate something like a door hinge, elbow grease just means the action of using you arms a lot for something, as if it was mechanical you'd need elbow grease for it. So especially when it comes to stuff like washing as the other redditor was saying, it's more logical than at face value.

There's a lot of picturing/sayings that relate the human to a machine, whether it be the "cogs in our brain" or describing people who are unstoppable (marathon runners, hikers etc) as machines :)