If I cast Thrill of the Hunt on my only unit at a battlefield that I have control of on my opponent's turn, do I score a point?

To determine if you score a point in this scenario, we must look at the rules for Control and Scoring (Rules 185-187, 464-466).

The Ruling

No, you will not score a point. Because you never lose control of the battlefield during the resolution of Thrill of the Hunt, re-entering the unit does not trigger a "Conquer" event.

Sequence of Events & Reasoning

1. Initial State: You control a battlefield.
2. Casting Thrill of the Hunt: When you play Thrill of the Hunt, the unit is banished and then played back to the battlefield. Per the Rules FAQ Clarification for Rule 187.4.c, the played unit is considered pending on the chain during this resolution.
3. Control Status: Because the unit is pending on the chain during the spell's resolution, you never actually lose control of the battlefield.
4. Re-establishing Control: Since you never lost control of the battlefield, placing the unit back at the site does not constitute "gaining control."
5. Conquer Requirement: Per Rule 464.1, a "Conquer" (which awards a score) only occurs if you gain control of a battlefield. Since you maintained continuous control throughout the spell's resolution, no new "Conquer" event occurs.

Key Nuances

• FAQ 9519 vs. Your Scenario: FAQ 9519 addresses a situation where control is actually lost and re-established. In your scenario, the specific interaction of Thrill of the Hunt maintains your control, meaning the requirements for a "Conquer" are not met.
• Conquer Requirement: A "Conquer" is a specific game event that triggers only when you seize control of a battlefield. Since you never relinquished control during the spell's effect, this re-entry does not constitute a new "Conquer" event (FAQ 8068).

Summary: You do not score a point because you did not lose and re-gain control of the battlefield; your control remained continuous throughout the resolution of the spell.

Note: I could not fully verify this ruling against the retrieved rules.