Let’s start with Exodus 20:13 and Deuteronomy 5:17—"You shall not kill." There are numerous other Scripture passages that condemn killing, such as Genesis 9:5—"For your lifeblood I will surely require a reckoning: of every beast I will require it and of man"—or Ezekiel 23:7—"Keep far from false charge, and do not slay the innocent and the righteous, for I will not acquit the wicked."
In the New Testament, Christ takes this command even further when he says, "You have heard it said to men of old, ‘You shall not kill; and whoever kills shall be liable to judgment.’ But I say to you that everyone who is angry with his brother shall be liable to judgment; whoever insults his brother shall be liable to the council, and whoever says, ‘You fool!’ shall be liable to the hell of fire."
But perhaps your friends are not convinced that these passages apply to taking one’s own life. If that is the case, ask them to consider Matthew 22:39, where Christ says that you must love your neighbor as yourself. If loving yourself means that you could kill yourself, then it would follow that you could do the same to your neighbor, and clearly that is condemned.
Instead of encouraging his wife in her desires for suicide, your friend must encourage his wife to live and convince her that he wants her to live and needs her to live. To do otherwise could have eternal consequences on them both.