<<<<That's very generous. And I agree, if you're inviting someone to a movie or restaurant meal. But a cruise and airfare obviously involve a lot more money>>>>>
Whether you're inviting someone to a movie or on a cruise, manners are manners -- it's not related to dollars and cents. I believe that if you invite, you pay. If you can't afford a steak dinner, then you eat hamburgers. If you can't afford a designer outfit, then you do without or buy something less expensive. If you cannot afford to pay for the child, then don't invite her.
Incidentally, I would not offer money (other than spending money) if my child were invited on such a trip. I'd assume that the family wouldn't offer if they hadn't considered the cost. Of course, I also wouldn't accept the offer (unless it were from an aunt/uncle or grandparents). I'd feel overly indebted to the other family and obligated to invite that child on my next vacation, which might not be in line with our family's plans!
Here's a slightly different skew to the situation: WHY are you inviting this child? I suspect it's not because you want to do something nice for her. Would you send her on the cruise alone just for her own enjoyment? Probably not. I suspect it's because you want your child to have a same-aged companion on vacation. Thus, the child is doing you a service by coming along and being that companion. Yes, I won't argue that the child is going to benefit too, but consider your real reasons for taking this child before you ask mom and dad for money!