God's purpose: his settled, prior plan to save, set in place before time and carried out in Christ
This is the word Paul reaches for when he wants to name the plan behind salvation. It does not mean a passing wish or a hope that things turn out well. It means a purpose already formed and firmly in place, the kind of fixed intention a king has when he sets a policy or a person has when their mind is made up. The same word was used in everyday Greek for a king's settled aim and for a person's resolve, and Paul lifts that ordinary sense and applies it to God: God has a plan, and that plan stands.
The senses below separate the two owners of such a purpose. Most often it is God's own purpose, the reason Paul can say that everything works together for good for those called according to it. Once or twice it is a person's resolve, the same firmness of mind turned toward how someone lives. The thread that holds them together is the idea of a purpose set beforehand and held to.
When Paul speaks of God's purpose he is making a large claim, and Christians have long disagreed about exactly how large. Some read it as God's eternal choice of particular people; others as God's single plan for his people and the world, reaching its goal in Jesus; others as God's sweeping claim to rescue a world in trouble. The disputes below lay out where the disagreement lies. What no one disputes is that for Paul this purpose belongs to God, was formed before time, and is now out in the open in Christ.