I think he was either an ancestor of Rama and perhaps it is also one of Rama's names. And what does it mean?
According to Brahma Purana, Chapter name: Vaivasvata Manu’s Children, Kakutstha was the son of Vikukshi a.k.a. Sasada. Vikukshi was the son of Ikshvaku.
But after Ikshvaku died, Vikukshi returned to his father’s kingdom and began to rule there. This was the kingdom of Ayodhya. One of Vikukshi’s sons was Kakutstha, and Rama of Ramayana fame was born in this line.
Rama is known as Kaakutstha means descendant of Kakutstha.
Kakutstha other names were Paranjaya or Puranjaya and Indravaha. He is known as Kakutstha because he fought a war seated on the shoulder (kakut) of Indra who was in form of bull. It is mentioned in Vishnu Purana, Book 4, Chapter 2.
Vikukshi had a very brave son called Paranjaya. The devas and the asuras once fought a terrifble war and the devas could not defeat the asuras. The devas prayed to Vishnu and Vishnu told them that he would be born on earth as Paranjaya. Under Paranjaya’s leadership, the devas would be able to defeat the asuras. So the gods came to paranjaya and asked him to be their leader in this war. Paranjaya agreed to do this only if he could fight te war perched on Indra’s shoulders. Indra adopted the form of a bull. Paranjaya fought the war seated on the bull. The demons wre defeated. But from the word kakut, which means shoulder. Paranjaya hereafter came to be known as Kakutstha.