There is nothing specific about the words. They are just a trigger phrase/word to get Hulk's attention, like "en guard" or "Avengers Assemble". Whedon said:
What about the words she says?
“Hey, big guy. Sun’s gettin’ real low.” I actually added that later in the game, I think something basically to get his attention and to have a phrase that he knows, this is about to start. I wanted to do as little with talking as possible, because it’s all going on there [points to his eyes], which is amazing, because two of those are not real — they were created by ILM, and yet they are so full of life. And I hate to say it, but he’s dead sexy as the Hulk.
As to how Black Widow and Hulk worked this out, it happens sometime between the first and second movie. Not much has been said, aside from Natasha and Banner having flirted and worked with each other in the time between movies. Their "connection".
When did the idea of a romance between Natasha and Bruce Banner begin percolating in your mind?
You know, we went there. I was trying to trace it back, because I honestly don’t remember before. We talked a little about the chemistry they had, just together in their scenes, and then I think it started with the lullaby. Like, “If you have the Hulk on the team, you know what would be cool is if Natasha’s the one who could talk him down.” And then it started to tick in my head, like, “There’s actually a truth there about who they are and how they connect that’s now very interesting,” and it built from there. Oddly enough, the lullaby itself is one of the last scenes that we finished because of the effects. And when we finished it, we were like, “This is way more romantic and way more physical than it was when it was post-viz, or a mark on a suit on a platform. It suddenly became that the romance was way more front and center in it than I expected.