I feel like I have been led to believe that Yamaguchi Sensei was the successor to Miyagi Sensei. I don't know why I have that in my head. But if he only studied for five years or less, then it would be improbable that Miyagi Sensei would make him his successor, especially if there were students in Okinawa who studied longer, including Yagi Sensei, who I have never heard of.
There was no official successor named By Miyagi Sensei. It is speculated that the closest thing Miyagi had to a senior student was Shinzato, Jinan, but he died during WWII. Shinzato may have been responsible for naming Goju-ryu. He was demonstrating his karate in Japan and when he was asked to name his style, which here-to-now had never had an identifying name other than Higashionna's karate or Miyagi karate, he reportedly said Goju-ryu or Hard Soft Way.
The senior students of Miyagi at his death included:
- Higa, Seiko
- Miyazato, Ei'ichi
- Yagi, Meitoku
- Toguchi, Seikichi
Higa was somewhat of an 'in between' man. He was only 10 years younger than Miyagi himself and thus he was old enough to have studied with Higashionna, Kanryo who was Miyagi's own teacher. After Higashionna passed away, Higa began to study with Miyagi, arguably Higashionna's senior student. I've always wondered why Higa was not acclaimed the head of Goju-ryu after Miyagi's death, but I guess we'll never know the full story as those who do know either are dead themselves or they aren't talking.
Miyazato generally was acknowledged as the leader of Okinawan Goju-ryu after Miyagi died. Whatever that means since it's obvious that Goju-ryu was never as high cohesive and organized as some other martial arts were. (Note: My teacher was a student of Miyazato, so I am naturally biased here.) Miyazato probably spent the most time with Miyagi Sensei out of his students (other than the deceased Shinzato) and he was one of the few that actually practiced karate with Miyagi post-war. Understandably, most Okinawans had other things on their minds than karate, like eating and surviving. Miyazato was also a Kodokan Judo 7th dan during his lifetime which is impressive in of itself.
Yagi was acknowledged as a kata expert among his peers. He was the first chairman of the Karate-do Goju-ryu Association established on Okinawa after Miyagi's death. Yagi created some kata unique to his lineage. There was an article about them in the Journal of Asian Martial Arts a few years ago. They are designed such that one forms the matched pair to the other. I'd like to learn all of the Yagi kata some day.
Toguchi started as a student of Higa's and later moved to Miyagi's dojo. He did a lot to spread Okinawan Goju-ryu teaching for a time in Japan before ultimately emigrating to Canada. His Shoreikan version of Goju is particularly popular in frozen tundra as a result.

Toguchi also created a few new kata as well as the so-called kiso kumite and bunkai sets used in his lineage.
Unrelated topic, but how much of the Kyokushin Kai curriculum is based on Goju Ryu? I understand that Oyama Sensei studied Shotokan and Goju Ryu as his main karate influences. I wonder if Oyama Sensei was the one who brought the Taikyoku kata to Yamaguchi Sensei.
Well, they do practice some Naha kata like Geikisai Dai Ichi and Dai Ni, Saifa, Seiunchin, Seipai, Seiryu, and Tensho. Oyama is said to have stated that all you need to know about his martial arts is expressed in Tensho, a creation of Miyagi, Chojun.
<shrugs> Kyokushin people perform these kata pretty differently than Okinawan Goju-ryu stylists. I'd say the system favors more the Shotokan influence than the Goju-ryu side, which isn't surprising at all given the dominance of the Taikyoku and Pinan kata in their kyu-grade syllabi.