To discover the available means of persuasion.
Exigence and audience are the primary building blocks of a rhetorical situation, in which a person is compelled to communicate with an audience. The range of possibilities are endless, from hitting one's self in the thumb with a hammer and crying out an expletive to the President taking the Oath of Office and delivering the inaugural address.
While figuring out what to say might be rather easy for the poor fellow who hit his thumb, other rhetorical situations require varying degrees of thought before we communicate. We must figure out what to say to achieve our desired goal. And this is the role of the first canon of rhetoric: invention.
A rhetorical situation demands that we discover:
Suppose, however, that you're called on to speak at a neighborhood political candidates' forum or write a letter to the editor about a civic issue. While a certain amount of natural rhetorical skill will be present (scholar George Kennedy says rhetoric acts like an instinct), wouldn't it be better for achieving your goal to have on hand a system for generating proper and effective material?
The process of invention, however, is not rigid. There is no set or proper way to employ it. The art of rhetoric requires each rhetor to acknowledge the fluid and contingent nature of human affairs. What works today might not work tomorrow. What works with one audience likely won't work with another.
Quite often an exigence involves a disagreement. The invention strategy of stasis theory provides a system for discovering the roots of the disagreement so that they may be addressed.
Don't we always know the source of our disagreements? Sadly, we do not. Much of the punditry and uncivil discourse of our culture is based on misunderstandings (willful and otherwise) of the sources of disagreement. The rhetor who would dig more deeply into issues might consider answering these questions before writing or speaking:
You might recognize the questions of stasis theory as similar to the basic critical questions we learn in school. The process of critical thinking is a process of invention. To discover what we think is to discover what we (might) have to say.