First, I'm sorry to hear about your friend's family situation - this is definitely tragic and I hope that at least medically a good resolution can be found.
I'm afraid that if every attorney has said that there is no hope after reviewing all of the facts, it sounds like you do have a clear answer on this already - just a very unfortunate one. These waivers are very difficult to get, and must be based on hardship to a citizen spouse or parent - not on hardship to a citizen child no matter how severe such a hardship might be.
Now, I can see making the argument that the child's illness, combined with the mother's illness which would make it difficult both to care for the child and to work to support the child, combine to create hardship for the citizen mother. If this can be documented with letters/reports from doctors, it would look to me as if maybe there is a case there based on just what is in your e-mail - but I hesitate to contradict other lawyers who may have access to more facts about the case than I. Further, a waiver case is expensive - not just legal fees but expert opinions - and this might be an issue, especially when there is no guarantee (or even a very high chance) of success.
I wish I could offer more hope.
I agree that this is a very tough case and the chances of getting the waiver are little, but if they don't do the waiver the chances are reduced to zero. The worst thing they can do is doing nothing. Immigration officers have a lot o prosecutorial discretion and if a good humanitarian case is made they can grant the waiver. I also believe there's nothing impossible for God and He can move anybody's heart.