In 5.10.6 of ISO 17025, it reads that:"When a calibration has been subcontracted, the laboratory performing the work shall issue the calibration certificate to the contracting laboratory."

My customer (the subcontracting lab) wants our lab (the lab performing the work) to put their customer's name and address on the certificates we issue for certain jobs.

In the section quoted above, does the word "issue" mean that I have to put my direct customer's name and address on the certification and cannot put the final customer's information? Or, does iit mean I just have to physically present the document to them and not to the final customer?

Simple answer: your customer is the lab that sent you the work. Their name is what goes on the certificate you issue.

They in turn should issue a cover cert (or similar method) that has their customer's name, with your cert as an attachment.

Hershal