The document was signed by nine councillors but five others and the mayor hadn't signed.
Victoria University is yet to disclose its intentions after Education Minister Chris Hipkins declined the name change proposal.
Deputy mayor Jill Day said it was for the university to decide whether it would try to progress with its name change and she did not need to be involved at this point.
Brian Dawson said every individual councillor was welcome to have a position on the name change.
But he did not think it was the role of council to have an official position.
"It's not something I'm prepared to go to war over but I don't particularly support it as a former Victoria student, but at the end of the day it's actually not a council decision."
Councillor Simon Marsh said the council should not force its opinion on the university over the issue.
There were plenty of other important decisions which needed the council's attention, he said.
"We shouldn't be looking at other people's business and unnecessarily poking our nose into it."
Councillor Peter Gilberd said a name in itself wasn't critical and the functioning of an organisation was much more important.
"I like Victoria as a name. I'm a Victoria graduate, I spent probably 20 years of my life there but I'm not going to get hung up on a name."
Councillor Simon Woolf said the name change decision rested with central government.
He said the notice of motion was not the correct mechanism to try to influence change.
"Victoria University's name change options are not council's core business, and while important, should not take our time, and be a distraction to the priorities we have on our plate currently."
Andy Foster is the councillor who moved the notice of motion.
He said they had already stuck their nose in the issue because Lester gave his support at a mayoral forum without consulting his elected colleagues first.
Foster said the council's view was important because one of the university's arguments for the name change was to be closer to the city and region.
A spokeswoman said there was no update on whether Victoria would challenge the Education Minister's decision.
The university council's next meeting is on March 25.
Depending on the university's next move, the notice of motion could be on Wellington City Council's table for debate on March 27.