If the design is performed effectively, the output will be a product
a.that meets customer requirement and satisfaction.
b.that has very minimal engineering changes post release.
c.that has very few or no field failures.
d.that is maintainable and has afforable life cycle cost.
and some more,depending on the intent of the design.
You may notice most of the metrics that can be derived from a to e are post release of product.
For metrics to monitor during the design process,deployment of Quality Function Deployment provides a very good framework.
For Project Management,metrics like Earned Value, Planned Value, Schedule Variance, Cost Variance are calculated. Based on these inputs Schedule Performance indicator and Cost Performance indicators are calculated and monitored.Projects Risks can be measured and monitored using attributes like Severity,occurence - prior,post mitigation.
You may want to refer to PMI handbook and ANSI/ISO/ASQ Q10006-Quality Management guidelines to quality in project management.
Procurement metrics in general are Quality, cost ,delivery, service.Some organizations have even better metrics encompassing TQRDC (HP model).
There are many useful references in the Quality field regarding development of performance metrics.If you are in an early stage of identification metrics, you may want to review Goal/Question/Metric Paradigm (one of the good reference).