Er, thoracic outlet syndrome is HUGE jump from a simple spinal nerve root or peripheral nerve impingement. TOS is a complicated condition that includes both neural and vascular symptoms and typically involves multiple impingement sites (cervical spine, scalenes, space anterior to the humeral head, and under pec minor). It could be from any number of things, particularly neck position, humeral head position, tricep muscle quality and mobility, etc. If he gets it more or if it starts bothering him at rest, tell him to see a orthopedist initially and then get a prescription for physical therapy. Nerve entrapment is something we treat regularly and have a lot of good results with. It may have to do with his neck, shoulder, elbow mobility, muscle length/mobility, postural control, form on the bench, etc. All of these things can be addressed by a good PT. Hope that helps.
Just for a quick clinical suggestion, I'd say it's either ulnar nerve impingement (due to tricep activation with poorly mobile triceps) or excess anterior glide of his humerus causing brachial plexus impingement (due to tight pec major and minor). Easy to fix for good with stretches, exercise, postural re-education, joint mobilizations, etc.