When did "give it up for" replace "let's have a big hand for"? It seems to have happened only a few years ago, and the switch seems to have happened quickly and completely. "Give it up for" strikes me as a very odd expression, although I'll admit that "a big hand for" is odd, too.
Maybe I just wasn't paying attention. That often happens.
Some day, no one will fully appreciate the clever title of that wonderful old comedy Western, A Big Hand for the Little Lady.