It sounds like your son may have a history of previous ankle sprains. A history of ankle sprains increases the risk of sprain recurrence. Shoe wear does not seem to make a difference.
Studies of athletes wearing high-top versus low top basketball shoes do not show a significant reduction of ankle injuries in one group over another. High-tops with inflatable chambers were also tested. They weren’t any more protective than other shoe types.
External ankle supports or braces have been shown to prevent ankle sprains. It’s not clear yet if the type of splint makes a difference. Ankle stirrup, semi-rigid bracing, elastic bandages, and lace up braces are available. Elastic wraps are less supportive than the more rigid supports. Some studies show faster results using a combination of elastic bandages and bracing.
Splinting is less expensive than the cost of taping for an entire sports season. But it does not appear to prevent ankle sprains in athletes who have already had three or more sprains.
Chronic sprains lead to ankle instability from loss of strength and impaired joint proprioception (sense of position). A physical therapy rehab program of exercises to restore normal ankle function is essential.
Preventive measures may not be helpful for athletes who have sprained the same ankle more than three times. Ankle instability from repeated sprains may require reconstructive surgery. If your son has a history of chronic, recurring ankle sprains, and he not been evaluated by an orthopedic surgeon, then a preseason exam may be a good idea.