There are a number of methods available in the literature. For example, Assessment of lean manufacturing effect on business performance using Bayesian Belief Networks-http://www.sciencedirect.com/science/article/pii/S0957417415002468
For a bunch of articles on this topic, please investigate these papers.
Truly lean management should have set of objectives and characteristics: elimination of waste, smooth operation flow, high level of efficiency, quality assurance. Realizing the lean objectives, lean should be supported on suitable supply chain management practices e.g. inventory minimization, higher resources utilization rate, information spreading trough the network, just-in-time practices, and shorter lead times. Measuring the leanness need to identify the major dimensions and indicators of lean as blow:
1. Quality: includes (product/service quality; quality of information sharing; quality of the partnership; customer reject rate; in plant defect fallow rate).
2. Cost: includes (production cost; eliminating non-value-added activities; savings in quality inspection costs and better integration of design efforts; cost per operating hour; New product flexibility).
3. Lead time : includes (reductions in vendor lead time; manufacturing cycle time, lead time from the order placing to the delivery).
I do not know if I have answered the question properly. Let me know and I will try to do it better.
All the above are great answers. I would like to add, that for each of the dimensions, weave in the organisation's perspective as to what they expected to achieve vis-a-vis what they actually achieved.
For example, on the cost dimension (building upon the previous answer), some of the parameters are:
Reduction in production cost; Elimination of non-value added activities; Savings in quality inspection costs; Higher integration of design efforts; Reduction in cost per operating hour; Greater new product flexibility.
For each parameter, gather what percentage benefit the organization expected to achieve as a result of lean management practices over a given period. Gather what percentage the organization actually achieved vis-a-vis their expectation.
This will help you isolate the benefit accruing from lean management alone, and separate it from other company initiatives. It also takes care of the fact that different organizations would be implementing lean at different levels of effectiveness.
The process of achieving organizational performance through lean management is similar to that of human body shredding extra fat to become lean, healthy, fit and active. The aim of lean management is to eliminate waste in every area of organization. Goal is continuous improvement through reduction in human effort, material, inventory, time, space; and improvement of customer satisfaction through improved quality & lower cost. Therefore the effect of lean management on organizational performance may be measured by measuring the achievements in above mentioned areas.
You might be willing to have a look at PhD thesis "Be lean to be resilient: Setting capabilities for turbulent times" recently defended by my colleague Dr. Seyoum Eshetu Birkie. The thesis is exactly about the interplay between the lean production, operational resilience and organizational performance. It a compilation thesis based on his papers published in established international journals, and the cover essay might be available online.