He first arrived on Monday last week.
Penguin colony environment team leader Henry Elsom said the seal had been going back and forth between Bushy Beach and the colony.
The colony has been referring to the seal as Rambo, but the Waitaki District Council has been calling him Big Jeff.
The name Big Jeff comes from a video of the seal posted to the council’s Facebook page by Buggy Robot Gallery owner Martin Horspool.
Mr Horspool said he was "really chuffed" to hear it had caught on.
"I used to live with a guy named Big Jeff who only had one finger and was very large.
"He’s passed away now ... I think he would be pleased his memory lived on in the form of an aquatic mammal."
The video was based on a show he watched growing up in the United Kingdom called Animal Magic, where the presenter would do silly voices and name all the creatures.
He also named a fur seal Myfanwy.
A council spokesman said it was always willing to listening to suggestions, which is why it adopted the name.
"We did not ask for the seal's identification."
Rūnanga o Moeraki upoko David Higgins said nobody had naming rights over individual animals, but the species had the traditional name ihu koropuku in te reo.
People were free to call the seal whatever they liked.
He suggested a few different names for the creature, including Bruce, Graham, Christopher and Simeon Brown.
