AMs have agreed that the National Assembly for Wales should change its name.
The assembly's presiding officer Elin Jones has told AMs that there are several possible choices.
Welsh Parliament has been proposed previously but one AM said the name Senedd should be considered.
But UKIP AM Gareth Bennett said some organisations with Welsh-only names were only known in the "Welsh-speaking colony" of Pontcanna in Cardiff.
First Minister Carwyn Jones said some would find his comments offensive.
No new name was agreed at an assembly debate to discuss the issue on Tuesday, but the presiding officer and assembly commission will now consider the implications of such a change.
AMs unanimously approved, without a vote, a motion which said the assembly "should change its name to reflect its constitutional status as a national parliament".
The assembly will soon get the power under the Wales Bill, which is passing through the UK parliament, to rename the National Assembly of Wales with agreement of two thirds of the chamber, with calls for the name change to reflect its growing powers.
The Welsh Government had planned a vote calling for the assembly to be known as the Welsh Parliament unofficially before a legal change could be made, but the vote was pulled.
Ms Jones said the next step "will be to consult soon on what the name should be".
"There are several possibilities and many associated terms which would stem from that choice," she said.
"The name should continue to inspire confidence and pride among the people of Wales."
She added she will notify the assembly of the next steps at the "outset" of the autumn term.
Bethan Jenkins, Plaid Cymru AM for South Wales West, has proposed that the Welsh word Senedd, which is used to refer to the assembly building, is used to refer to the institution itself.
"Let us be self-confident enough to discard our mental baggage and take this opportunity to be unique by adopting an official Welsh name for our Senedd," she told the debate.
She said some AMs wish to retain the name National Assembly, but added Senedd is already used widely on the ground.
Welsh Conservative leader Andrew RT Davies, who suggested the Assembly was renamed to the Welsh Parliament back in 2012, said: "I couldn't see why we couldn't have Senedd and Parliament, because obviously for people there is a complete understanding of the word Parliament, in the way UK democracy works."
"Over time I think the word Senedd could be readily taken on board," he added.
UKIP group leader Neil Hamilton wondered whether the public at large would regard the debate as "pretentious navel gazing".
"As an when the Wales Bill passes and tax raising powers are devolved to this institution there is, of course, a serious argument for calling ourselves a Parliament," he said.
He said he was personally attracted to the idea of the Senedd as a name. "I wonder whether that would make us Senators", he added.
But Gareth Bennett, UKIP AM for South Wales Central, called for the term Senedd to be "kicked into the long grass" and said it would distance the assembly "from the majority of the people of Wales".
"Bethan is from a Welsh speaking background, or a bilingual background, and in her social circle Senedd may be a well used term," Mr Bennett said, to laughs and derision from Ms Jenkins.
"In the Wetherspoons pub in Canton few people would know what I was talking about if I started talking about the Senedd, and I can tell you that from experience."
Referring to a list of organisations with Welsh-only names, including the Urdd and Chwarae Teg, that Ms Jenkins had read out Mr Bennett said: "The problem is most people outside the Welsh speaking colony of Pontcanna, most people in Cardiff don't know what these things are."
First Minister Carwyn Jones said: "If someone stood up in this chamber and described people in Ceredigion or in Gwynedd who are English speakers as a colony there would be uproar."
"It makes it sound as if people in Wales who speak a certain language don't belong in our capital city, and that is a wholly wrong remark," he said, adding it would be "highly offensive to a number of people" although he wasn't sure Mr Bennett meant it that way.