There's that whole "You can lead a horse to water..." saying, and it's true. You can't force someone to accept your help, no matter how hard you try. You can, however, try to explain yourself and your reasoning for wanting to help, and hope that they decide to agree with you.
I can't say, since I don't really know the people involved, but by your account, your friend sounds like she's being incredibly self-destructive right now; not only is she refusing much-needed aid out of stubborn pride, she is also attacking you for your offer of help. Explain why you felt you needed to try to sneakily give her money, and how her lashing out at a helping hand isn't going to solve her problems any more than sitting and suffering through them alone.
Explain yourself to her; tell her how you care too much about them to watch them go through this hardship, when you actively want and are able to help them, comfortably... that accepting the help of others isn't a sign of weakness. If she's the sort of person I suspect she is, she would do that sort of thing for someone else in a heartbeat, so why does it make sense to refuse it the other way around? It's unfair to people who care about her, and to herself and her child, to be this self-destructive.
Listen Truean, I know you really don't want to have to consider this, but if she's letting the kid starve you need to call child services.
....
Fuck.
I've repeatedly been asked why I post questions (business, ethical, etc) "like that on a video game forum." My inherent incompleteness, doesn't allow me to see everything. Here, my pride, my attachment ... obscures my sight and objectivity. Now, there's too much I don't know and at stake in too little time for my personal failings.
So yeah, sit down with her--like Solifuge said--and see exactly what's wrong, and if she won't, scout situation with P.I. to see if they're at least getting food somewhere, call in outside objective attorney or social worker I trust.... If she has no food, I wonder how long it will be until she has no apartment....
If she's just really hit hard financial times, I'll soften the blow--which will hit whether she faces it or not--as much as possible. If that's the case, I can make sure she is given shelter and food. I really hope the cynical side of me is wrong, and that this isn't drugs or something. I could never conceive of her doing that, which is exactly why I can't be objective here. I really don't like where this is headed....
Thank you for your time and consideration,
Truean