Well, the neoreactionaries claim that strong monarchies will be more stable, and less subject to needing to satisfy the fickle whims of the population. There is some validity to at least part of the argument: long-term projects may do better in dictatorships. Look for example at the US space program: there's an argument that part of why it has stalled is that each President, desiring to have a long-lasting legacy, makes major changes to the program's long-term goals, so every few years a lot of work in progress is scrapped. Certainly that's happened with the last three Presidents. And the only President whose project really stayed beyond his office was JFK, who had the convenience of being a martyr and having a VP who then cared a lot about the space program's goals.

However, the more general notion that monarchies are more stable as a whole is empirically false, as discussed in the anti-reaction FAQ.

What I suspect may be happening here is a general love for what is seen as old, from when things were better. Neoreaction may have as its core motivation a combination of cynicism for the modern with romanticism about the past.

If you do read any of the pro-reaction stuff linked to by K (or the steelman of reaction by Yvain) I suggest you then read Yvain's anti-reaction FAQ which provides a large amount of actual data.