Laws vary from state to state and I am unaware of any Ohio laws governing lengths of blades on knives. However, it would be highly unlikely that you son would be criminally charged with carrying a dangerous weapon, unless he made statements that he carries the knife to protect himself, (that would then get him charged with carrying a concealed or dangerous weapon). Lastly, however, he is probably going to be in some trouble with the school as most schools have zero tolerance regarding the carrying of knives and other such items on school property.
A deadly weapon is an object, when used, has the potential of causing death.
I think your question is more along the line of "When is a knife considered a weapon vs. just a pocket knife of inconsequence?"
There is no clear definition of what length a knife blade needs to be in order to be defined as a weapon. It is left to the discretion of the police officer. Most officers I know tend to go by their badge length (about 3 inches). If the blade is bigger than their police badge it is a weapon and charge the person with CCW (carrying a concealed weapon). If it is smaller than their badge it is just an object in the pocket - unless used as a weapon.
Your son may be charged with carrying a weapon on school property, depending on the police and the prosecutor in conjunction with the school's attitude. You should call the police officer handling the matter to find out if a report was forwarded to the prosecutor's office. If so, you may wish to start looking for an attorney who practices in the juvenile court to defend your son in this matter, as this could be charged as a felony which may have future consequences.