Yes, because having the wrong name on his card may delay you are your son's tax refund in the future.
I know I shouldn't have waited so long, but my son's name was changed 5 years ago. Will this be an issue in verifying his identity?
No, there will not be an issue as long as you have an identity document with your son's prior name on it.
What all do I need ot do to change his name on his Social Security card?