Somebody who says "I can make my own destiny," is endorsing what psychological concept?

Free Will.

Explanation:

Free will is the idea that behaviour is a reflection of choices made by the individual.

Free will is the notion that overall all things, we are the product of our choices. Contrary to deterministic views, some psychologists recognize that free will is a cognitive process by which decisions are taken, and a course of action is set. It involves an action based on purpose and a primary of human psychological function, for which choosing always occurs either automatic or as a habit that deliberately was planned over time.

There is a debate for which extent will behaviour result as conflicting motives, or if people will determine based on external or internal factors.

In any circumstance, we have a degree of choice that makes us free to determine our behaviour, so we say we are self-determined.