You've hit it head on - it continues because she gives in. That's why he only does it with her; that's where it works. It will continue until she gains the determination to put her foot down and stick to it. Once she does, it will continue for a month or so (the longer the behavior has been permitted, the harder it will be to extinguish) and even get worse for a bit, but she has to stay consistant or it will come right back! And she doesn't want to let this behavior (basically maniuplating parents to get around the rules) continue much longer, because he's going to school soon and she needs to be able to get him up and ready on time, to do homework, etc. And definitely it has to stop before he's a teen (long time out, yet, and he won't be whining then, it'll be some other way of manipulating).
However, she will only be able to change this if she is WILLING to. The only thing you can do is offer her support and suggestions (if she's willing to listen). I would gently remind her that it is her RESPONSIBILITY as a mom to teach her son there are limits. Let her know you know what it's like - you've been there! - and that it gets much better afterwards. I would tell her there are times when she is going to have to just be "mean" - grounding or whatever, but again, she's got to do it, things will get better, and she'll enjoy being a mom much more when he learns that pushing her buttons will just put him in timeout. Be there for hugs and support and a listening ear, no matter what she decides. If she ends up saying "oh well, it's not a big deal, I don't care" don't lecture her (she knows it's not ok, but can't deal with it at the moment). If she wants help, offer your suggestions.