Maybe I'm missing something here, but this seems way too unnecessarily difficult. I was able to use the "add field to analysis layer" tool to add a field that did show up in my output service area lines, but it was always completely null when trying several different input options in the tool. I eventually used "Select Data" tool with the 'facilities' option and then copy features to export a point feature class with the "STOP_ID" field in my input point layer. Then I joined the output lines using Facilities.ObjectID = ServiceAreaLines.FacilityID. Then I used "Calculate Field" to finally get the STOP_ID onto the output service areas lines.
Why can't users just specify a series of fields in the "Add locations" tool and those fields be present (and populated) in all of the Network Analyst output!?!?!?
I think the documentation on this issue needs to be more fleshed out. Sample geoprocessing models would also be helpful.