To have a referendum I think you either need a resolution passed by elected members at a Full Council meeting or in response to a petition signed by at least 5% of local residents.
According to Wikipedia "As of May 2014, there have been 51 referendums on the question of changing executive arrangements to a model with an elected mayor. Referendums are triggered by council resolution, local petition or central government intervention. Of these, 16 have resulted in the establishment of a new mayoralty and 35 have been rejected by voters. Average "yes" vote is 45%. Typical turnout is around 30%, but has been as low as 10% and as high as 64%. The turnout is higher when the referendum coincides with another vote, such as an election.
There have been four referendums on the question of removing the post of elected mayor. Two mayoral posts have been disestablished following a vote and two retained".
Directly elected Mayors can be imposed without a referendum and are very much in the news lately with Government plans for more regional and strategic Mayors.
Hat tip my former UNISON NEC colleague Lesley Discombe. The picture caption is of a famous Torbay resident who I am sure would have fancied himself as Mayor.