Assuming you have verified that this is all true (because people do create bogus profiles and bogus charities), I'd focus my time on finding out if there are already services that exist to connect this mom and other parents to the generosity of others. Start with the hospital social workers and sew what they already do to provide respite care, parking vouchers, toys, activities, etc. for sick kids. Then start a neighborhood or school-wide effort to support that charity. You will get more participation from your friends because they will know it's a real organization or charity they are supporting. You will get better suggests that are relevant to the actual kids being serviced, instead of just guessing at what they need. Individual circumstances vary so much, you can't possibly identify or anticipate all of them. So work with the people who are already in the trenches, and you will make sure your efforts are multiplied.
I've seen FB pages set up for kids getting treatment - it protects their privacy to some degree. One kid was not able to continue his animal therapy for some reason, so they set up a page letting people post "greetings" from their own dogs with a message like, "Hi Billy, hope you enjoy this picture while you are getting chemo. My name is Rufus and I'm a Boston terrier, I like to play fetch and take naps." That sort of thing - priceless, warm encouragement and entertainment without loading a kid up with 20 Nintendos that he might not even like or need.