As I could read the majority of people who disagrees with people having this right, chose this view because of their belief in god as our creator, hence his choice about when and how you should die.
What if suicide was God's will? What if his way of making someone achieve his purpose in here was by letting this person go by his own will? It was mentioned the movie The Sea Inside somewhere in this thread. Those who know the story will agree with me that dead was pretty meaningful (the movie was based on true events, I must say).
I read some of you guys claiming that the suicide is a sign of the sickness of the person. Well, maybe it's a sign of the sickness of the world.
For me almost everyone should have this right, if it comes to that. But not for all the cases. People should face the consequences of certain acts. A guy who enters a school shooting at dozens of children shouldn't have the right to escape by killing himself.