I've been thinking about this since you posted.
Legally, according to the present legal system, they do have the right to insist.
Morally, I'm not so sure they should. I don't think the impact would be lessened simply because it's a medical procedure...
I think it would depend on the child too. I definitely feel that kids who know what they're talking about and are informed should be allowed to be part of their own medical care. If they want to refuse a certain kind of treatment, etc., they should be allowed if they are well informed of all the implications.
I think I'm leaning towards the rights of the kid, but only if they know exactly what they're talking about.