There are several different "none of these" options that have been implemented. Many of them are only implemented in private elections (as distinct from public elections; e.g. elections within unions or student elections or internal elections in political parties).
One is "Re-Open Nominations" (aka RON). If elected, then the nominations have to be re-opened and then the election re-run. Sometimes, those nominated the first time are ineligible to be nominated again.
A second possible result is to create a casual vacancy (as if the elected person had died in office), which is then filled by the normal mechanism for filling a vacancy. If that normal mechanism is a by-election (US: special election) then this is effectively the same as RON*. But it may not be; for instance US Senators are elected, but vacancies are filled with appointment by the state's governor; the President is elected, but a vacancy is filled by the Vice-President; VP vacancies are filled by Presidential appointment and confirmation by both Houses of Congress.
A third possible result is that the office will sit unfilled for one term and its powers cannot be exercised for that term. This is more common with elections to committees or parliaments than for executive office - the parliament is just one member short for a few years.
In a primary election, the result of NOTA winning is usually that that party will not contest the election. In primaries that are closed, especially ones that are narrowly closed (like US caucuses, or French-style primaries that are only open to dues-paying members), the option not to contest the election may be used to concentrate the party's resources on particular elections.
* There may be technical differences between a by-election and a rerun of the normal election. For example, in the UK, the expenditure limits are higher for by-elections than for re-run elections. In the US, different rules may apply for primary elections for a special election than apply for a general election, and not all states have rules for a rerun general election at all (reruns can apply if, for example, a candidate dies too late for their name to be removed from ballots, as well as for a victory by NOTA/RON).