So as I wrote, once he gets a new passport from home, bring both passports, enter on the new passport using the old one. Then go to the Phuket office to transfer stamps when he returns to Thailand. And make sure that he is stamped in on his new passport correctly as his extension (with "Non-RE" as visa class on entry stamp, until November 3), before leaving the passport control when he enters Thailand.
Actually, there is no Non-O for the reason of attending school. Also, no Non-O multiple entry (still valid) for reasons other than being a spouse to a Thai (there are a few very specific exceptions, though).
Now Doubtful your son has an actual multiple-entry Non-O visa. If he is studying at a real school, he could be on 1-year extension with a multiple re-entry permit. Or up to 3 months extension (incremental) if he is studying at a private school, such as a language school.
Check his current visa or stamp and make sure what he is on now, at least whether he has an entry stamp directly from a visa, or an extension (for whatever period).
He can enter Thailand being stamped on his new passport, using his current visa/extension on his old passport. But visa/extension cannot be transferred at entry point. He needs to go to the local office that issued his extension to transfer stamps once he is back to Thailand. Don't forget getting re-entry permit before leaving Thailand, and don't forget to bring his old passport back to Thailand.
Only at the immigration office that issued your current one-year extension. If yours was issued at Chaengwattana, you must go to the Chaengwattana office.