Locate Caleb McCaffery and take possession of his hat. This can be achieved by either killing him or with a high enough speech skill, he can be convinced to give it to you.
Return to the casino and as a reward for the hat you get another key, the Garret's gratitude and some good room company.
After the quest Debt Collector, talk to Francine Garret she will tell you a man by the name of Caleb McCaffery has stolen caps and has fled to the Strip. So she asks you to go and kill him and to retrieve his hat as proof.
Go to the strip to find Caleb near the outside of Gomorrah and initiate conversation with him. You have 3 choices, one involving no dialogue.
With a high enough speech skill, you can simply ask for his hat, and he will give it up. Turn in the hat to Francine to complete the quest and gain his room key as a reward. Taking the peaceful solution leaves Caleb alive, but reveals that he is bald. It's unknown whether he will stay indefinitely.
Tell him that you are a collector for the Garrets and ask him to hand over the hat. He'll respond negatively and start combat. He carries nothing but a shotgun for a weapon and a duster for armor, so killing him should be relatively easy once you've earned a few levels. Take his hat off his body and return to Francine to claim your reward.
Alternatively, you can simply sneak attack him without initiating dialogue. If your attack is strong enough, you can potentially kill him instantly and avoid a fight. Loot the hat from his corpse afterward.