Introduction
The Dune books have some interesting psychological research embedded into them that (finally) after 50 years might be discussed and used (like Frank Herbert predicted). However it is also likely that the perversion that is the Dune movies (which are visually stunning, but in many ways lacking story-wise) will just allow people to 'grok' what Dune is about while ignoring what it's really about. As usual, the story is done by a naturalist (Good guys beautiful, bad guys ugly). Whether the movies about the second book will ever materialize is questionable.
There are many fans of the Dune books like Jeff Bezos or Tim O'Reilly, both Silicon Valley titants. I highly recommend reading Tim O'Reilly's book on Frank Herbert https://www.oreilly.com/tim/herbert/ to better understand some of the topics the books cover.
Give as few orders as possible
Right before facing the trial of riding a sand worm the following text can be found:
More of his men were stirring out of their tents. Guards were coming in from the rims. Everything around him moved smoothly in the ancient routine that required no orders. "Give as few orders as possible" his father told him...once...long ago. "Once you're given orders on a subject, you must always give orders on that subject"
A lot of the above stinks of John Boyd and some of his conclusions on military organisation. However, Dune precedes Boyd's writing by a decade.
"We care for our own"
The Atreides training - "We care for our own!" - it held like a core of native rock in them, Halleck noted.
It is notable that an organisation without explicit rules can only function if there is no in-fighting. In other words, the staff must be trained to take care of their neighbour. This is very different from the hierarchical management the Harkonnen use, where the Harkonnen can never sleep at night because they are always playing a shady game. Most organisations operate in the Harkonnen fashion.
Only loyalty buys loyalty
This is a big principle and it showcases the importance the Atreides put on getting the right people and keeping them. This is contrasted by the way Harkonnen carelessly use people.
Perhaps the most important lesson is that it's not possible to operate in the Atreides way if you scroll Tinder-for-Work (LinkedIn), which most management and HR does. Actual loyalty is required to receive loyalty. Hiring a bunch of LinkedIn people will ruin the company culture very fast.
"The people must learn how well I govern them"
We mustn't run short of filmbase", the Duke said. "Else, how could we flood village and city with our information? The people must learn how well I govern them. How would they know if we didn't tell them?
This is one of the more cynical conversations between the Duke Leto Atreides and his son Paul. But it shows a certain romanticism i.e. the importance of a consensus reality overlayed over the naturalistic facts. In a certain sense, it is important to counteract disinformation efforts by the enemy. The Atreides cared about the population under them.
The bravura
This is broken across the books, and it is not easy to find all the quotes. The earliest is right before the jom gabbar test:
Now there was a man who appreciated the power of bravura - even i death, the Reverend Mother thought.
The opinions of the ruled matter, and sometimes it is very necessary to be unselfish and do something that does not benefit you in any way. The old duke Atreides fought in the corrida and died. The power of a spectacle shouldn't be underestimated. It forges a consensus reality and a joined tau among the participants. This is not a 'panem et circenses' approach.
Conclusion
The two core principles are to give as few orders as possible and to also ensure people take care of each other. It's not possible to self-organize or operate in this manner without both.
There are more Atreides principles peppered throughout the books (and their shortcomings). I might be missing quite a few of them.