Bob Beckel's return should help improve Fox News ratings. He was well liked before he left and possibly those who liked him well enough, may have followed him to another news program. If those viewers return with him, then no doubt the ratings will improve. If the ratings go back up, Fox would do well to do all in their power to keep him this time.
No, I don't think that Bob Beckel's return will improve ratings for Fox News. I personally know a lot of people who will not listen to Fox News because they see it as a biased news outlet with questionable sources and reporters. Just changing the co-host isn't enough to save Fox News.
Fox is Fox and it appeals to a fairly specific audience, just as it turns off other segments of the news watching public. I don't think Beckel's return will make much of a difference either way. Those who like Fox and it's way of presenting things will continue to watch, and those who don't won't.
He was not popular when he was on the network, and this is not likely to change not that he is back. He did have some fans, and is amazing at negotiating a contract, but the network is trying to save itself in a era when all news outlets are sure to die.