Would a cool mist humidifier help my child's sore throat?

Yes: Yes, especially if he has a post nasal drip.

Maybe,maybe not: These help most when the environment is very dry, like in winder, when the heater pulls the humidity down below 20%.Humidifiers & vaporizers can help bring up the humidity up where it is not further drying the kids airway when they are sick. If the basic humidity the kid is in runs 30-40%, the humidifier has less positive benefit.