The collapse of the island's heavily indebted banks led to demands for change after accusations of cronyism between the political elite and business.
The referendum is non-binding but backers of change hope that politicians will find it hard to ignore even though parliament is responsible for adopting a new constitution and the main opposition party has said it opposes proposed changes. Saturday's referendum asked voters six questions, including whether people wanted a new constitution which has been drawn up by a specially-appointed panel of 25 citizens to be the basis for a review of the basic law.
With two-thirds of votes counted on Sunday, 66 percent had answered "yes" to that question. Turnout was 49 percent of the island's more than 235,000 eligible voters, broadcaster RUV said.
"This is a very clear conclusion for parliament. The majority of voters want changes in all the topics asked about in the vote," said Thorolfur Matthiasson, an economist at the University of Iceland.
He noted 80 percent had voted to declare all non-privately owned natural resources as "national property".
Fishing accounts for about 7 percent of the economy with fishing rights currently farmed out under a system of quotas which critics say have benefited a select few. Backers of the system say it has led to sound management of fish stocks.
"There will be pressure to change the fishing quota system because people want a bigger share of income from fishing and other natural resources," said Matthiasson.
Control of the island's natural resources remains a sensitive issue. Plans by a Chinese tycoon to buy rural land were blocked by the government last year. He is to lease the land instead.
In 2011, a Canadian company also faced protests - led by singer Bjork - and eventually agreed to reduce its stake in a geothermal power company.
The draft constitution includes provisions to allow 10 percent of voters to call their own referendums. It also sets a limit on the terms a president can serve to three from the current unlimited terms.
The draft constitution was drawn up after deliberations by the 25 members of the council and after about 3,600 comments and 370 suggestions were made to the council's website. The council also used Facebook and Twitter to communicate with the public.