A child is having trouble with a project at school and feels like he can't do it. The teacher comes along and sees that the child is struggling. "Just do your best." The teacher says, to give the child some motivation to keep trying.
I think that's what people assume that phrase means. But really think about what "Do the best you can" means. It doesn't mean "Just get something done" or "try hard" It means do the best you can. If you told Leonardo da Vinci to do the best he can, he would produce the Mona Lisa. If you told Einstein to do the best he can, he would produce the theory of relativity. Do the best you can means produce the highest quality result that you are possibly able to do.
Part of me finds it interesting how a phrase which means something when you look at the words individually, but the context of it means almost the exact opposite. Similar to Nimrod. The other part of me (I only have two parts) thinks it's a waste of a very powerful sentence.
Do the best you can.