Some reasons are common and some differ from country to country. Since other answers cover mostly common reasons, I add some reasons that are special to one or several of them. First I'd note that I don't consider all examples mentioned below as authoritarian governments. I only say if one consider it authoritarian, it has stability because...
Religion: e.g Iran. In Iran avid fans of Ayatollah Ali Khamenei are mostly because of his religious position. Largest percentage of population to attend a funeral was for Ayatollah Ruhollah Khomeini, The founder of the Islamic Republic of Iran.
Severe repression: e.g. Saudi Arabia.
In some countries there are elections that one person is the winner until dying. e.g. Azerbaijan. In this case people are not unanimous if the government is a kind of dictatorship.
not considered as authoritarian e.g. UK. It is widely accepted (true? or false?) that Queen has not notable role in the power.
Support of foreigners: e.g. Bahrain, One of the most sever dictatorships. Bahrain's started large protests against its rulers in early 2011. A month later the king requested security assistance from Saudi Arabia and ... . Also Bahrain's dictator has support of US and UK.
No good replacement: After many efforts the government overthrown; what then? It takes decades to reach a democracy like what in the US, which itself is not a good government system. people are socket puppet of owners of money and power. When they are tired of republicans they vote to democrats to change (e.g. vote for Obama). Then they don't see good changes. Again they vote to republicans and,...
Note that countries are subject to a number of cases above; e.g. the case "Severe repression" comes true also for Bahrain.
See also my answer to this questionthis question.
