She offered a night. He wants a whole week.
Yes, but the original scenario was for something like a blizzard, where the streets presumably would be cleared the next day. No one was envisioning the worst hurricane that the East Coast got since 1938 or so.
I don't think that he was wrong in asking regarding a week, and I think it's kind of hair-splitting for PPs to assert that it was really not an emergency for him. Just because "I got to work okay, it took a bit longer, that was all" does not mean that this was universal for every other person working in the area.
A week is a lot to ask, I think we all understand that here. And OP is not rude for saying no. But it would naive to figure that there would be no consequences for saying no.
We don't know that there will be consequences, either. I have a close friend who suffers from some serious mental and emotional conditions. And there have been times (although not in circumstances this severe) when she's been unable to do something she promised, or forgotten something important, or something along those lines. And when she realizes, she apologizes, we work around it as best we can, and we go on from there. Yes, sometimes I get frustrated or upset, but mostly, I'm upset with her illness, not with her. I know she's doing the best she can, and she's very open with me about when things are especially bad and she needs a little extra understanding.
It sounds like the OP's friend is aware of her illness, and of how it affects her during a rough patch. OP, I think, if you can, you should talk candidly with your friend and tell him that a week just isn't going to work for you, and you're really sorry. If a night or two could be made to work, then offer that, or see what other options might be workable.
I do think that there is a bit of "snap out of it" in this thread, and that's never helpful. Mental illness is as real and devastating as any other serious, chronic illness.