There have been very few times when a 1st round KO was considered worthy of a repeat, but it has happened. Sometimes it's because a sanctioning body forces it. Other times it can be due to an odd finish, or controversy. However, as in the first fight, usually anything can happen the second time. Let's look at a few examples:
Sonny Liston vs. Floyd Patterson
Not sure why Patterson wanted this rematch. After being dropped 2 minutes into his first fight with Liston, he either couldn't, or wouldn't, beat the count. However, the heavyweight title had changed hands, and Liston had pretty much already cleaned out the division just getting there. So, nine months later, they did it again. It happened the same way, except Patterson got up twice... lasting a full 4 seconds longer than he had, originally.
Juan Manuel Lopez vs. Daniel Ponce DeLeon
This rematch was by no means immediate. Lopez actually won his first world title and stormed onto the world scene with a 1 round drubbing of formerly iron chinned DeLeon. He went on a tear, winning another belt as well, before a string of consecutive KOs ended his young career. However, in that bad streak, his one big win was a rematch with DeLeon. This time, he himself was dropped early, but still Juanma managed to finish Ponce in round 2. Therefore, DeLeon has the dubious honor of being responsible for Juanma's first and last big wins.
Rocky Gannon vs. Dominick Carter
The lone avenging story in our list... these two USA Cruiserweights were competing for a minor belt on a televised card, when they took turns knocking each other silly. The referee mistook a bad trip, as a stagger from Gannon, and stopped the fight. The controversy led to the rematch, and although the fireworks were similar, the difference in talent and conditioning showed up as early as round 3. Finally, Carter was rescued (right before he quit) in round 5.
Joe Calzaghe vs. Mario Veit
This we can blame on a bogus sanctioning body, because there was nothing controversial about the first fight. Calzaghe battered Veit to the floor twice in round 1 and finished him. However, that was Veit's first loss in 31 fights, so it was no surprise when 15 fights later, he re-earned a shot at Calzaghe. Veit may have been able to beat his fellow Germans in Ottke and Beyer, but promotional problems stopped those fights, and he was no match for the Welsh hall of famer. This time, Veit took 6 rounds to be dropped twice and finished. He never challenged for a world title again.