Wow, I love mamasource. What a great set of responses you've gotten. Now you have the hard problem of having to make the decision of how to handle it. Good luck to you.
I'll offer my opinion here just to be complete. I think it's a horrible situation. I agree that you should have a back up plan for all events you schedule with her. Simply tell your kids that since sometimes your friends aren't able to make it, you can still do something fun. My kids like back up plans and they like knowing it'll work out even if the play date is questionable.
I do worry something bigger is wrong with your friend. She may be super sensitive because she can't explain it (like a social anxiety) or because she is embarrassed. However, I think it is perfectly fine to tip toe through the subject with her. Simply putting a positive spin on it will help. For example, ask if her daughter was disappointed about the cancelling. Perhaps you can then delve deeper into how the little girl handled the issues in the past. This allows you to indirectly bring up that it is habitual. Is the daughter asking to reschedule? Perhaps you can ponder out loud how your children would feel about missing their party. See how she reacts. As the kids get older, their school friends will want to know what's up and kids can be mean. If the girl is in school, you can ask if the little girl got teased at school or if the kids were sympathetic. Perhaps the mom needs to think about the fact that it has a social impact on her daughter. Indirect questions always seem like the best way to go when the person is touchy. I have a sister like this and you have to gently talk about all the things around the "giant purple elephant" so she'll bring it up or she'll shut down and say I just don't understand. They should teach courses in this!!!
Also, ask if your friend is feeling better and is there anything you can do to help. Perhaps you can ask, do you think it is allergies? Ha, joke that maybe it's morning sickness and see if she defends her symptoms. If she appears to still be not feeling well, ask if she needs you to bring by some dinner to help out. That would make her feel guilty if she isn't really sick and I would guess she'd back pedal a little bit.
If she is really shut down, quietly tell her (like it's a secret) that you're worried about her and want to help if you can. Perhaps she just needs to know you're available to confide in. Actually, offer to her that if she ever needs someone to talk to in confidence, that you can be that person and that you won't judge her. Everyone needs to let off some steam sometimes and it is helpful to know it won't make you "hate the husband", "think they are crazy", "think they are a bad parent", etc. Maybe she is depressed. Complaining about her husband may lead her to realize that it isn't her husband that is the problem... or maybe it is.
I've told a friend before that she is welcome to call me when ever she is needing to vent because I know that sometimes you need to release that stress to better look at the situation. I told her I won't judge the situation because I know I'm only hearing one side and it will be tainted with emotion. So she knows she can confide in me. When I ask questions about her situation, I try to be very careful not to feed the frustration while supporting her, listening to her (which can be so theraputic) and possibly offering more positive feedback.
Also, I love the idea of offering to take the little girl to the event for your friend next time so she can get some rest and feel better. I love the idea of suggesting to have the party yourself and then celebrating in honor of the child if she doesn't make it. Not like you have so much free time and money with 3 kids :) , but it is a nice idea. We hosted a mini make up party for a friend who was in the hospital during her birthday party and had to cancel. We only had her, her sister and mother over but we hung streamers and blew up balloons and had fun plates... and my kids loved it as much as she did. If she had cancelled, we would have still had fun. Perhaps doing something smaller would appeal to your friend if it is social anxiety. Also, if it is a financial issue, you could suggest to your friend that you've heard about birthday parties where they order a bunch of cheese pizzas (loved by kids and relatively cheap) and take them to a big park with a playground. Bring some balls, bubbles and a streamers (cheap at dollar stores) and the kids will have a blast. Help the kids to play tag or simon says or tie streamers to a stick and run around making it wave in the wind. The party is fun without much expense. Give the kids a few rolls of streamers or toilet paper (make sure they clean it up) and tell them to decorate themselves. Kids love to run wild and use their imagination.
So, now that I've given you more than a mouthful of advice... good luck. I hope you are able to strengthen your relationship with this friend. You know, if all else fails and she gets upset or acts negatively... be blunt and tell her, a friend cares and wants to help. I'm trying to be a good friend. If I didn't care about my friends I'd find superficial friends and talk about he weather or a new recipe. She may not get it at first, but who wouldn't stew on that for a while and realize that you really care?
Good luck. I hope to read a wonderful success story soon.
You are a wonderful person to care enough to try to help. Try to talk kindly about the mother and daughter to your children so they don't harbor your frustration (and maybe anger) too. Kids pick up on so much...
Liz
p.s. Also, next time she makes plans with you, I think it would be very acceptable for you to say something like, "great, we'd love to do that and if you aren't able to make it we'll go anyway" or "... we'll just go to -XYZ- which is close by." That brings it to her attention (nicely) that you don't feel confident they'll come while removing some of the stress she may feel about HAVING to make the event. (sorry, I'm so wordy today :)