The Dandelion Project said the mayor would be "symbolic" but the election would be a study in democracy.
Marc Winn, founder of the project, said he wants to see if island-wide electronic voting can work.
Chief Minister Jonathan Le Tocq said it was an interesting concept and he was in favour of engaging more people in politics.
However, he said a mayor would not solve the question of how Guernsey should be governed.
"We are not going to please everyone, there is not a utopia out there but there is a journey and the more people on the journey the better it is for everyone," he said.
Guernsey's parliament, the States of Deliberation, is made up of 47 representatives, elected as people's deputies, in multi-member districts, and two representatives of the States of Alderney.
Mr Winn said the successful candidate would have no power, but it could show that island-wide voting was feasible.
He said: "We could be testing an electronic voting system for island wide voting, which the population wants but the government can't work out how to achieve."
The Dandelion Project was launched with the goal of making Guernsey "the best place to live on earth by 2020".
Mr Winn said it had been set up to bring islanders with a range of skills together to work to improve conditions for everyone.
A community group is to set up an election for an honorary mayor of Guernsey.