I am going to go against the grain a bit here as I think YABabitU
You offer her a lift because you have a car to do so ... she doesn't have a car so she cannot offer a lift to you. She is also not in a position to offer someone else's car to you as it is not her "favour" to offer.
When you fell through as a lift she just looked for a different lift, which you were also free to do and she was under no obligation to arrange for you. She can't offer a lift to you when she is not the one driving as it is not her place to offer someone else's car/time/petrol to collect you. (I would always be happy to pick up friends/family if they needed a lift but would not appreciate them then offering me as a lift to other people without at least consulting me first)
She could have asked you if she should ask if her son/friend to pick you up too but if you didn't ask her she may not have thought of it (I know I possibly wouldn't under those circumstances), there was nothing to stop you saying "Oh, if X says they can give you a lift could you ask if they can pick me up too/let me know so I can ask them too?", and she is under no obligation to organise your transport when you made no effort yourself (did you think to phone said mutual friend to ask for a lift?)
You made no indication to her that the only reason you weren't going was because of the transport, because you didn't say anything about trying to find alternate transport. She may have assumed that since the car was out of action you'd decided on a quiet night in instead of faffing with lifts and going out, especially as you hadn't indicated otherwise.
I used to do a pub quiz with friends every week but it was a casual standing arrangement - if someone couldn't come one week then it was no problem, we'd see them next time. Sometimes I couldn't make it because my babysitter fell through on the day ... I could have found another babysitter if I really wanted to go but some weeks I wasn't that bothered and I wasn't too sad to miss the quiz as a one off. I would have assumed your friend thought this was how you were feeling, you'd have liked to go but weren't bothered about missing this one, especially since you made no effort to find an alternative way to get there.