Customer Focus is Central to Lean Thinking

Agile UX vs Lean UX – How they’re different and why it matters for UX designers by Anders Ramsay

Good design is good design, regardless of if we are living in an Agile or Lean or [insert new term] universe. The only real difference when adopting these methods is in the How.
...
So, while there certainly is a lot that UX designers can learn from Agile thinking on software delivery (such as to automate everything that can be automated), the real pay-off for UX practitioners is in the collaboration part.
...
Lean UX, on the other hand, is really a reference to Lean Startup a la Eric Ries (and Steve Blank, basically the godfather of Lean Startup, and creator of concepts like Customer Development and GOOBing), which extends Agile ideas to include methods for ensuring that there in fact is a market for the product you are doing such a good job designing and shipping early and often.

Lean is also very concerned with customer value. From that perspective usability is directly relevant and fits perfectly in the lean context. Lean focused on the ideas of usability for customers in the world of physical products. Usability of software fits with this perfectly.

The focus of reducing waste in lean context is to look at the "value stream" and eliminate what doesn't add value to the customer. The primacy of user is there (though some applying lean don't understand this).

Dr. Deming said the customer is the most important part of the production line. He understood the customer (end user) had to be the focus of your efforts to improve. Toyota took Deming's ideas and created the Toyota Production System (Shoichiro Toyoda, Honorary Chairman and director of Toyota: "There is not a day I don't think about what Dr. Deming meant to us. Deming is the core of our management.". Lean manufacturing is the name given to the practices of Toyota as Jim Womack and Dan Jones as they researched the company (and wrote The Machine That Changed the World.